Chapter 1. Measurement 1.What is Physics? 2. Measuring Things

Chapter 1. Measurement 1.What is Physics? 2. Measuring Things
3. The International System of Units 4. Length 5. Time 6. Mass 7. Changing Units 8.Calculations with Uncertain Quantities

What is Physics? Physics is the study of the basic components of the universe and their interactions. Theories of physics have to be verified by the experimental measurements.

Measurement A scientific measurement requires:
(1) the definition of the physical quantity (2) the units. The value of a physical quantity is actually the product of a number and a unit . The precision of the measurement result is determined by procedures used to measure them.

Basic Measurements in the Study of Motion
Length: Our “How far?” question involves being able to measure the distance between two points. Time: To answer the question, “How long did it take?” Mass: Mass is a measure of “amount of stuff.”

The Système International (SI) of units
The SI, or metric system of units is the internationally accepted system of units for measurement in all of the sciences, including physics. The SI consists of base units and derived units: (1) The set of base units comprises an irreducible set of units for measuring all physical variables (2) The derived units can be expressed in terms of the base units

The SI Base Units

Time: One second is the duration of 9
Time: One second is the duration of × 109 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

Length: One meter is the distance traveled by light in a vacuum in a time interval of 1/ of a second

Mass: One kilogram is the mass of this thing
(a platinum-iridium cylinder of height=diameter=39 mm) Atomic mass units (u)

Scientific Notation All Physics quantities should be written as scientific notation, which employs powers of 10. The Order of magnitude of a number is the power of ten when the number is expressed in scientific notation

Example Determine the order of magnitude of the following numbers:
(a) A=2.3×104, (b) B=7.8×105.

Changing Units In chain-link conversion, we multiply the original measurement by one or more conversion factors. A conversion factor is defined as a ratio of units that is equal to 1. For example, because 1 mile and 1.61 kilometers are identical distances, we have:

EXERCISE 1 (a) Explain why it is correct to write min/60 s = 1, but it is not correct to write 1/60 = 1. (b) Use the relevant conversion factors and the method of chain-link conversions to calculate how many seconds there are in a day .

EXERCISE 2 The cran is a British volume unit for freshly caught herrings: 1 cran= liters (L) of fish, about 750 herrings. Suppose that, to be cleared through customs in Saudi Arabia, a shipment of 1255 crans must be declared in terms of cubic covidos, where the covido is an Arabic unit of length: 1 covido=48.26 cm . What is the required declaration?

Density The density ρ of a material is the mass per unit volume:

Calculations with Uncertain Quantities
Significant Figures: Read the number from left to right, and count the first nonzero digit and all the digits (zero or not) to the right of it as significant. Significant figures and decimal places are different The most right digit gives the absolute precision, which tells you explicitly the smallest scale division of the measurement. Relative Precision is the ratio of absolute precision over the physics quantity.

EXERCISE 3 Determine the number of significant figures, absolute precision, relative precision in each of the following numbers: (a) 27 meters, (b) 27 cows, (c) second, (d) – × 10–11 coulombs, (e) 5280 ft/mi.

EXERCISE 4 Suppose you measure a time to the nearest 1/100 of a second and get a value of 1.78 s. (a) What is the absolute precision of your measurement? (b) What is the relative precision of your measurement?

A Simple Rule for Reporting Significant Figures in a Calculated Result
Multiplying and Dividing: When multiplying or dividing numbers, the relative precision of the result cannot exceed that of the least precise number used Addition and Subtraction: When adding or subtracting, you line up the decimal points before you add or subtract. This means that it's the absolute precision of the least precise number that limits the precision of the sum or the difference. Data that are known exactly should not be included when deciding which of the original data has the fewest significant figures. Only the final result at the end of your calculation should be rounded using the simple rule. Intermediate results should never be rounded.

EXERCISE 5 Perform the following calculations and express the answers to the correct number of significant figures. (a) Multiply 3.4 by (b) Add 99.3 and 98.7. (c) Subtract 98.7 from 99.3. (d) Evaluate the cos(3°). (e) If five railroad track segments have an average length of meters, what is the total length of these five rails when they lie end to end?

Chapter 2 Motion Along a Straight Line
2-0. Mathematical Concept 2.1. What is Physics? 2.2. Motion 2.3. Position and Displacement 2.4. Average Velocity and Average Speed 2.5. Instantaneous Velocity and Speed 2.6. Acceleration 2.7. Constant Acceleration: A Special Case 2.8. Another Look at Constant Acceleration 2.9. Free-Fall Acceleration 2.10. Graphical Integration in Motion Analysis

Trigonometry Example 1 Using Trigonometric Functions On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is =50.0°, as Figure 1.6 shows. Determine the height of the building.

Trigonometric Functions
PYTHAGOREAN THEOREM

Example 1 Using Trigonometric Functions
On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is =50.0°, as Figure 1.6 shows. Determine the height of the building.

What is the location of downtown Wilmington?

Defining a Coordinate System
One-dimensional coordinate system consists of: a point of reference known as the origin (or zero point), a line that passes through the chosen origin called a coordinate axis, one direction along the coordinate axis, chosen as positive and the other direction as negative, and the units we use to measure a quantity

Scalars and Vectors A scalar quantity is one that can be described with a single number (including any units) giving its magnitude. A Vector must be described with both magnitude and direction. A vector can be represented by an arrow: The length of the arrow represents the magnitude (always positive) of the vector. The direction of the arrow represents the direction of the vector.

A component of a vector along an axis (one-dimension)
A UNIT VECTOR FOR A COORDINATE AXIS is a dimensionless vector that points in the direction along a coordinate axis that is chosen to be positive. A one-dimensional vector can be constructed by: Multiply the unit vector by the magnitude of the vector Multiply a sign: a positive sign if the vector points to the same direction of the unit vector; a negative sign if the vector points to the opposite direction of the unit vector. A component of a vector along an axis=sign × magnitude

Difference between vectors and scalars
The fundamental distinction between scalars and vectors is the characteristic of direction. Vectors have it, and scalars do not. Negative value of a scalar means how much it below zero; negative component of a vector means the direction of the vector points to a negative direction.

Check Your Understanding 1
Which of the following statements, if any, involves a vector? (a) I walked 2 miles along the beach. (b) I walked 2 miles due north along the beach. (c) I jumped off a cliff and hit the water traveling at 17 miles per hour. (d) I jumped off a cliff and hit the water traveling straight down at 17 miles per hour. (e) My bank account shows a negative balance of –25 dollars.

Motion The world, and everything in it, moves.
Kinematics: describes motion. Dynamics: deals with the causes of motion.

One-dimensional position vector
The magnitude of the position vector is a scalar that denotes the distance between the object and the origin. The direction of the position vector is positive when the object is located to the positive side of axis from the origin and negative when the object is located to the negative side of axis from the origin.

Displacement DISPLACEMENT is defined as the change of an object's position that occurs during a period of time. The displacement is a vector that points from an object’s initial position to its final position and has a magnitude that equals the shortest distance between the two positions. SI Unit of Displacement: meter (m)

Example 2: Determine the displacement in the following cases:
(a) A particle moves along a line from to (b) A particle moves from to (c) A particle starts at 5 m, moves to 2 m, and then returns to 5 m

EXAMPLE 3: Displacements
Three pairs of initial and final positions along an x axis represent the location of objects at two successive times: (pair 1) –3 m, +5 m; (pair 2) –3 m, –7 m; (pair 3) 7 m, –3 m. (a) Which pairs give a negative displacement? (b) Calculate the value of the displacement in each case using vector notation.

Velocity and Speed A student standing still with the back of her belt at a horizontal distance of 2.00 m to the left of a spot of the sidewalk designated as the origin.

A student starting to walk slowly
A student starting to walk slowly. The horizontal position of the back of her belt starts at a horizontal distance of 2.47 m to the left of a spot designated as the origin. She is speeding up for a few seconds and then slowing down.

Average Velocity x2 and x1 are components of the position vectors at the final and initial times, and angle brackets denotes the average of a quantity. SI Unit of Average Velocity: meter per second (m/s)

Example 4 The World’s Fastest Jet-Engine Car
Figure (a) shows that the car first travels from left to right and covers a distance of 1609 m (1 mile) in a time of s. Figure (b) shows that in the reverse direction, the car covers the same distance in s. From these data, determine the average velocity for each run.

Example 5: find the average velocity for the student motion represented by the graph shown in Fig. 2-9 between the times t1 = 1.0 s and t2 = 1.5 s.

Average Speed Average speed is defined as:

Check Your Understanding
A straight track is 1600 m in length. A runner begins at the starting line, runs due east for the full length of the track, turns around, and runs halfway back. The time for this run is five minutes. What is the runner’s average velocity, and what is his average speed?

EXAMPLE 6 You drive a beat-up pickup truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station. (a) What is your overall displacement from the beginning of your drive to your arrival at the station? (b) What is the time interval from the beginning of your drive to your arrival at the station? What is your average velocity from the beginning of your drive to your arrival at the station? Find it both numerically and graphically. Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min. What is your average speed from the beginning of your drive to your return to the truck with the gasoline?

Instantaneous Velocity and Speed
The instantaneous velocity of an object can be obtained by taking the slope of a graph of the position component vs. time at the point associated with that moment in time The instantaneous velocity can be obtained by taking a derivative with respect to time of the object's position. Instantaneous speed, which is typically called simply speed, is just the magnitude of the instantaneous velocity vector,

Example 7 The following equations give the position component, x(t), along the x axis of a particle's motion in four situations (in each equation, x is in meters, t is in seconds, and t > 0): (1) x = (3 m/s)t – (2 m); (2) x = (–4 m/s2)t2 – (2 m); (3) x = (–4 m/s2)t2; (4) x = –2 m. (a) In which situations is the velocity of the particle constant? (b) In which is the vector pointing in the negative x direction?

How to Describe Change of Velocity ?

Definition of Acceleration
SI Unit of Average Acceleration: meter per second squared (m/s2)

Instantaneous acceleration:

An object is accelerated even if all that changes is only the direction of its velocity and not its speed. It is important to realize that speeding up is not always associated with an acceleration that is positive. Likewise, slowing down is not always associated with an acceleration that is negative. The relative directions of an object's velocity and acceleration determine whether the object will speed up or slow down.

EXERCISE A cat moves along an x axis. What is the sign of its acceleration if it is moving (a) in the positive direction with increasing speed, (b) in the positive direction with decreasing speed, (c) in the negative direction with increasing speed, and (d) in the negative direction with decreasing speed?

EXAMPLE 7: Position and Motion
A particle's position on the x axis of Fig. 2-1 is given by with x in meters and t in seconds. (a) Find the particle's velocity function and acceleration function . (b) Is there ever a time when (c) Describe the particle's motion for ?

Constant Acceleration: A Special Case

Free-Fall Acceleration

Equations of Motion with Constant Acceleration

Example 8 A Falling Stone
A stone is dropped from rest from the top of a tall building, as Figure 2.17 indicates. After 3.00 s of free-fall, what is the velocity of the stone? what is the displacement y of the stone?

Example 9 An Accelerating Spacecraft
The spacecraft shown in Figure 2.14a is traveling with a velocity of m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing

Example 10 Spotting a police car, you brake your Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. What is that acceleration? (b) How much time is required for the given decrease in speed?

Graphical Integration in Motion Analysis

Conceptual Question A honeybee leaves the hive and travels 2 km before returning. Is the displacement for the trip the same as the distance traveled? If not, why not? Two buses depart from Chicago, one going to New York and one to San Francisco. Each bus travels at a speed of 30 m/s. Do they have equal velocities? Explain. One of the following statements is incorrect. (a) The car traveled around the track at a constant velocity. (b) The car traveled around the track at a constant speed. Which statement is incorrect and why?

At a given instant of time, a car and a truck are traveling side by side in adjacent lanes of a highway. The car has a greater velocity than the truck. Does the car necessarily have a greater acceleration? Explain. The average velocity for a trip has a positive value. Is it possible for the instantaneous velocity at any point during the trip to have a negative value? Justify your answer. An object moving with a constant acceleration can certainly slow down. But can an object ever come to a permanent halt if its acceleration truly remains constant? Explain.

Chapter 3. Vector 1. Adding Vectors Geometrically
2. Components of Vectors 3. Unit Vectors 4. Adding Vectors by Components 5. Multiplying Vectors

Adding Vectors Graphically
General procedure for adding two vectors graphically: (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of . General procedure for adding two vectors graphically: (1) On paper, sketch vector to some convenient scale and at the proper angle. (2) Sketch vector to the same scale, with its tail at the head of vector , again at the proper angle. (3) The vector sum is the vector that extends from the tail of to the head of .

Examples

Two important properties of vector additions
(1) Commutative law: (2) Associative law:

Subtraction

Two vectors, A and B, are added by means of vector addition to give a resultant vector R: R=A+B. The magnitudes of A and B are 3 and 8 m, but they can have any orientation. What is (a) the maximum possible value for the magnitude of R? (b) the minimum possible value for the magnitude of R?

Unit Vectors The unit vectors are dimensionless vectors that point in the direction along a coordinate axis that is chosen to be positive

How to describe a two-dimension vector?
Vector Components:The projection of a vector on an axis is called its component .

Properties of vector component
The vector components of the vector depend on the orientation of the axes used as a reference. A scalar is a mathematical quantity whose value does not depend on the orientation of a coordinate system. The magnitude of a vector is a true scalar since it does not change when the coordinate axis is rotated. However, the components of vector (Ax, Ay) and (Ax′, Ay′), are not scalars. It is possible for one of the components of a vector to be zero. This does not mean that the vector itself is zero, however. For a vector to be zero, every vector component must individually be zero. Two vectors are equal if, and only if, they have the same magnitude and direction

Example 1 Finding the Components of a Vector
A displacement vector r has a magnitude of r 175 m and points at an angle of 50.0° relative to the x axis in Figure. Find the x and y components of this vector.

Reconstructing a Vector from Components
; Magnitude: Direction:

Addition of Vectors by Means of Components
q =

Two vectors, A and B, have vector components that are shown (to the same scale) in the first row of drawings. Which vector R in the second row of drawings is the vector sum of A and B?

Example 2 The Component Method of Vector Addition
A jogger runs 145 m in a direction 20.0° east of north (displacement vector A) and then 105 m in a direction 35.0° south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacements.

Multiplying and Dividing a Vector by a Scalar

The Scalar Product of Vectors (dot product )
The dot product is a scalar. If the angle between two vectors is 0°, dot product is maximum If the angle between two vectors is 90°, dot product is zero

The commutative law

Example What is the angle between and ?

The Vector Product (cross product )
(3) Direction is determined by right-hand rule (1) Cross production is a vector (2) Magnitude is

Property of vector cross product
The order of the vector multiplication is important. If two vectors are parallel or anti-parallel, . If two vectors are perpendicular to each other , the magnitude of their cross product is maximum.

Sample Problem In Fig. 3-22, vector lies in the xy plane, has a magnitude of 18 units and points in a direction 250° from the +x direction. Also, vector has a magnitude of 12 units and points in the +z direction. What is the vector product ?

Sample Problem If and , what is ?

Chapter 4 Motion in Two and Three Dimensions
4.1. What is Physics? 4.2. Position and Displacement 4.3. Average Velocity and Instantaneous Velocity 4.4. Average Acceleration and Instantaneous Acceleration 4.5. Projectile Motion 4.6. Projectile Motion Analyzed 4.7. Uniform Circular Motion 4.8. Relative Motion in One Dimension 4.9. Relative Motion in Two Dimensions

What is Physics?

Position and Displacement
Position vector: Displacement :

EXAMPLE 1: Displacement
In Fig., the position vector for a particle is initially at and then later is What is the particle's displacement from to ?

Problem 2 A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by At t=15 s, what is the rabbit’s position vector in unit-vector notation and in magnitude-angle notation?

Average and Instantaneous Velocity
Instantaneous velocity is:

Particle’s Path vs Velocity
Displacement: The velocity vector The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

Problem 3 A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by At t=15 s, what is the rabbit’s velocity vector in unit-vector notation and in magnitude-angle notation?

Average and Instantaneous Acceleration
Average and Instantaneous Acceleration Average acceleration is Instantaneous acceleration is

Speed up or slow down If the velocity and acceleration components along a given axis have the same sign then they are in the same direction. In this case, the object will speed up. If the acceleration and velocity components have opposite signs, then they are in opposite directions. Under these conditions, the object will slow down.

Problem 4 A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit’s position as functions of time t (second) are given by At t=15 s, what is the rabbit’s acceleration vector in unit-vector notation and in magnitude-angle notation?

How to solve two-dimensional motion problem?
One ball is released from rest at the same instant that another ball is shot horizontally to the right The horizontal and vertical motions (at right angles to each other) are independent, and the path of such a motion can be found by combining its horizontal and vertical position components. By Galileo

Projectile Motion A particle moves in a vertical plane with some initial velocity but its acceleration is always the free-fall acceleration g, which is downward. Such a particle is called a projectile and its motion is called projectile motion.

Properties of Projectile Motion
The Horizontal Motion: no acceleration velocity vx remains unchanged from its initial value throughout the motion The horizontal range R is maximum for a launch angle of 45° The vertical Motion: Constant acceleration g velocity vy=0 at the highest point.

A projectile is fired into the air, and it follows the parabolic path shown in the drawing. There is no air resistance. At any instant, the projectile has a velocity v and an acceleration a. Which one or more of the drawings could not represent the directions for v and a at any point on the trajectory?

Example 4 A Falling Care Package
Figure shows an airplane moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m. The directions to the right and upward have been chosen as the positive directions. The plane releases a “care package” that falls to the ground along a curved trajectory. Ignoring air resistance, (a). determine the time required for the package to hit the ground. (b) find the speed of package B and the direction of the velocity vector just before package B hits the ground.

Example 5 The Height of a Kickoff
A placekicker kicks a football at an angle of θ=40.0o above the horizontal axis, as Figure shows. The initial speed of the ball is (a) Ignore air resistance and find the maximum height H that the ball attains. (b) Determine the time of flight between kickoff and landing. (c). Calculate the range R of the projectile.

UNIFORM CIRCULAR MOTION
Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path

Properties of UNIFORM CIRCULAR MOTION
Period of the motion T: is the time for a particle to go around a closed path exactly once has a special name. Average speed is : This number of revolutions in a given time is known as the frequency, f.

Example 6 A Tire-Balancing Machine
The wheel of a car has a radius of r=0.29 m and is being rotated at 830 revolutions per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving.

CENTRIPETAL ACCELERATION
Magnitude: The centripetal acceleration of an object moving with a speed v on a circular path of radius r has a magnitude ac given by Direction: The centripetal acceleration vector always points toward the center of the circle and continually changes direction as the object moves.

The car in the drawing is moving clockwise around a circular section of road at a constant speed. What are the directions of its velocity and acceleration at following positions? Specify your responses as north, east, south, or west. position 1 position 2

Example 7 The Effect of Radius on Centripetal Acceleration
The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as Figure illustrates. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. Express the answers as multiples of g=9.8 m/s2.

Relative Motion in One Dimension
Relative Motion in One Dimension The coordinate The velocity The acceleration

Relative Motion in Two Dimension
Relative Motion in Two Dimension The coordinate The velocity The acceleration

Sample Problem In Fig. 4-23a, a plane moves due east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity relative to the wind, with an airspeed (speed relative to the wind) of 215 km/h, directed at angle θ south of east. The wind has velocity vpG relative to the ground with speed of 65.0 km/h, directed 20.0° east of north. What is the magnitude of the velocity of the plane relative to the ground, and what is θ?

Chapter 5. Force and Motion I
5.1. What is Physics? 5.2. Newtonian Mechanics 5.3. Newton's First Law 5.4. Force 5.5. Mass 5.6. Newton's Second Law 5.7. Some Particular Forces 5.8. Newton's Third Law 5.9. Applying Newton's Laws

What Causes Acceleration?
Dynamics—the study of causes of motion. The central question in dynamics is: What causes a body to change its velocity or accelerate as it moves?

Newtonian Mechanics Newton’s laws fail in the following two circumstances: 1. When the speed of objects approaches (1% or more) the speed of light in vacuum (c = 8×108 m/s). In this case we must use Einstein’s special theory of relativity (1905) . 2. When the objects under study become very small (e.g., electrons, atoms, etc.). In this case we must use quantum mechanics (1926).

Force A force is a push or a pull. Force is a vector. All forces result from interaction. Contact forces: forces that arise from the physical contact between two objects. Noncontact forces: forces the two objects exert on one another even though they are not touching.

External forces include only the forces that the environment exerts on the object of interest.
Internal forces are forces that one part of an object exerts on another part of the object.

Combining Forces

Principle of superposition for forces
When two or more forces act on a body, we can find their net force or resultant force by adding the individual forces as vectors taking direction into account. Note: The net force involves the sum of external forces only (internal forces cancel each other).

Exercise 1 The figures that follow show overhead views of four situations in which two forces acting on the same cart along a frictionless track. Rank the situations according to the magnitudes of the net force on the cart, greatest first.

Newton’s FIRST LAW

Newton's First law: Consider a body on which no net force acts
Newton's First law: Consider a body on which no net force acts. If the body is at rest, it will remain at rest. If the body is moving, it will continue to moving with a constant velocity. “Net force” is crucial. Often, several forces act simultaneously on a body, and the net force is the vector sum of all of them An inertial reference frame is the one has zero acceleration. All newton’s laws are valid only in the inertia reference frames.

Mass The larger the mass, the harder is to cause its motion
Mass and weight are different concepts

DEFINITION OF INERTIA AND MASS
Inertia is the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line. The mass of an object is a quantitative measure of inertia. SI Unit of Inertia and Mass: kilogram (kg)

NEWTON’S SECOND LAW OF MOTION
When a net external force acts on an object of mass m, the acceleration a that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force. = SI Unit of Force: kg·m/s2 =newton (N) Only External forces are considered in the Newton’s second law.

Example 1 Pushing a Stalled Car
Two people are pushing a stalled car, as Figure 4.5a indicates. The mass of the car is 1850 kg. One person applies a force of 275 N to the car, while the other applies a force of 395 N. Both forces act in the same direction. A third force of 560 N also acts on the car, but in a direction opposite to that in which the people are pushing. This force arises because of friction and the extent to which the pavement opposes the motion of the tires. Find the acceleration of the car.

Example 2 Hauling a Trailer
A truck is hauling a trailer along a level road, as Figure 4.32a illustrates. The mass of the truck is m1=8500 kg and that of the trailer is m2= kg. The two move along the x axis with an acceleration of ax=0.78 m/s2. Ignoring the retarding forces of friction and air resistance, determine (a) the tension T in the horizontal drawbar between the trailer and the truck and (b) the force D that propels the truck forward.

Questions The net external force acting on an object is zero. Is it possible for the object to be traveling with a velocity that is not zero? If your answer is yes, state whether any conditions must be placed on the magnitude and direction of the velocity. If your answer is no, provide a reason for your answer. Is a net force being applied to an object when the object is moving downward (a) with a constant acceleration of 9.80 m/s2 and (b) with a constant velocity of 9.80 m/s? Explain. Newton’s second law indicates that when a net force acts on an object, it must accelerate. Does this mean that when two or more forces are applied to an object simultaneously, it must accelerate? Explain.

Newton's Third Law

Newton's Third Law If one object is exerting a force on a second object, then the second object is also exerting a force back on the first object. The two forces have exactly the same magnitude but act in opposite directions. Forces always exist in pairs. It is very important that we realize we are talking about two different forces acting on two different objects.

Question A father and his seven-year-old daughter are facing each other on ice skates. With their hands, they push off against one another. Compare the magnitudes of the pushing forces that they experience. Which one, if either, experiences the larger acceleration? Account for your answers.

EXAMPLE 3: Pushing Two Blocks
In Fig. 3-29a, a constant horizontal force of magnitude 20 N is applied to block A of mass 4.0 kg, which pushes against block B of mass 6.0 kg. The blocks slide over a frictionless surface, along an x axis. What is the acceleration of the blocks? What is the force acting on block B from block A

Example 4 The Accelerations Produced by Action and Reaction Forces
Suppose that the mass of the spacecraft in Figure 4.7 is mS= kg and that the mass of the astronaut is mA=92 kg. In addition, assume that the astronaut exerts a force of P=+36 N on the spacecraft. Find the accelerations of the spacecraft and the astronaut.

NEWTON’S LAW OF UNIVERSAL GRAVITATION

NEWTON’S LAW OF UNIVERSAL GRAVITATION
Every particle in the universe exerts an attractive force on every other particle. For two particles that have masses m1 and m2 and are separated by a distance r, the force that each exerts on the other is directed along the line joining the particles and has a magnitude given by: The symbol G denotes the universal gravitational constant, whose value is found experimentally to be

DEFINITION OF WEIGHT The weight of an object on or above the earth is the gravitational force that the earth exerts on the object. The weight always acts downward, toward the center of the earth. On or above another astronomical body, the weight is the gravitational force exerted on the object by that body. SI Unit of Weight: newton (N)

When the height of object H above the Earth is small

The Gravitational Acceleration Constant
When air resistance can be ignored and any object under only gravitational force will free fall with a constant acceleration:

RELATION BETWEEN MASS AND WEIGHT
Mass is an intrinsic property of matter and does not change as an object is moved from one location to another. Weight, in contrast, is the gravitational force acting on the object and can vary, depending on how far the object is above the earth’s surface or whether it is located near another body such as the moon.

Questions When a body is moved from sea level to the top of a mountain, what changes—the body’s mass, its weight, or both? Explain.

Questions The force of air resistance acts to oppose the motion of an object moving through the air. A ball is thrown upward and eventually returns to the ground. (a) As the ball moves upward, is the net force that acts on the ball greater than, less than, or equal to its weight? Justify your answer. (b) Repeat part (a) for the downward motion of the ball.

The Normal Force The normal force FN is one component of the force that a surface exerts on an object with which it is in contact—namely, the component that is perpendicular to the surface.

APPARENT WEIGHT

The Friction Force When the object moves or attempts to move along a surface, there is a component of the force that is parallel to the surface. This parallel force component is called the frictional force, or simply friction. It is always against the relative motion or the attempts of the motion between object and surface.

Tension Tension is the force exerted by a rope or a cable attached to an object Tension in a Nonaccelerating rope: the magnitude of tention is the same everywhere in the rope. An Accelerating rope: the magnitude of tension is not the same everywhere in the rope that has a mass; however, the magnitude of tension is the same everywhere in the rope that is massless.

For a massless string It is always directed along the rope.
It is always pulling the object. 3. It has the same value along the rope

Applying Newton's Laws Newton's Second Law:
It can be written as two (or three) component equations:

Equilibrium Applications of Newton's Laws of Motion
DEFINITION OF EQUILIBRIUM: An object is in equilibrium when it has zero acceleration.

EXAMPLE 5: Three Cords In Fig. 6-25a, a block B of mass M = 15 kg hangs by a cord from a knot K of mass mK, which hangs from a ceiling by means of two other cords. The cords have negligible mass, and the magnitude of the gravitational force on the knot is negligible compared to the gravitational force on the block. What are the tensions in the three cords?

Free-Body Diagrams Identify the object for which the motion is to be analyzed and represent it as a point. (2) Identify all the forces acting on the object and represent each force vector with an arrow. The tail of each force vector should be on the point. Draw the arrow in the direction of the force. Represent the relative magnitudes of the forces through the relative lengths of the arrows. (3) Label each force vector so that it is clear which force it represents.

Example 6 Replacing an Engine
An automobile engine has a weight W, whose magnitude is W=3150 N. This engine is being positioned above an engine compartment, as Figure 4.29a illustrates. To position the engine, a worker is using a rope. Find the tension T1 in the supporting cable and the tension T2 in the positioning rope.

Example 7 Equilibrium at Constant Velocity
A jet plane is flying with a constant speed along a straight line, at an angle of 30.0° above the horizontal, as Figure indicates. The plane has a weight W whose magnitude is W= N, and its engines provide a forward thrust T of magnitude T= N. In addition, the lift force L (directed perpendicular to the wings) and the force R of air resistance (directed opposite to the motion) act on the plane. Find L and R.

Non-quilibrium Applications of Newton's Laws of Motion
DEFINITION OF NONEQUILIBRIUM: An object is in nonequilibrium when it has non-zero acceleration.

Example 8 Applying Newton’s Second Law Using Components
A man is stranded on a raft (mass of man and raft=1300 kg), as shown in Figure a. By paddling, he causes an average force P of 17 N to be applied to the raft in a direction due east (the +x direction). The wind also exerts a force A on the raft. This force has a magnitude of 15 N and points 67° north of east. Ignoring any resistance from the water, find the x and y components of the raft’s acceleration.

Example 9 Towing a Supertanker
A supertanker of mass m=1.50×108 kg is being towed by two tugboats, as in Figure. The tensions in the towing cables apply the forces T1 and T2 at equal angles of 30.0° with respect to the tanker’s axis. In addition, the tanker’s engines produce a forward drive force D, whose magnitude is D=75.0×103 N. Moreover, the water applies an opposing force R, whose magnitude is R=40.0×103 N. The tanker moves forward with an acceleration that points along the tanker’s axis and has a magnitude of 2.00×10–3 m/s2. Find the magnitudes of the tensions T1 and T2.

Conceptual Questions According to Newton’s third law, when you push on an object, the object pushes back on you with an oppositely directed force of equal magnitude. If the object is a massive crate resting on the floor, it will probably not move. Some people think that the reason the crate does not move is that the two oppositely directed pushing forces cancel. Explain why this logic is faulty and why the crate does not move. A stone is thrown from the top of a cliff. As the stone falls, is it in equilibrium? Explain, ignoring air resistance. Can an object ever be in equilibrium if the object is acted on by only (a) a single nonzero force, (b) two forces that point in mutually perpendicular directions, and (c) two forces that point in directions that are not perpendicular? Account for your answers.

A circus performer hangs stationary from a rope
A circus performer hangs stationary from a rope. She then begins to climb upward by pulling herself up, hand over hand. When she starts climbing, is the tension in the rope less than, equal to, or greater than it is when she hangs stationary? Explain. A weight hangs from a ring at the middle of a rope, as the drawing illustrates. Can the person who is pulling on the right end of the rope ever make the rope perfectly horizontal? Explain your answer in terms of the forces that act on the ring.

Chapter 6 Force and Motion II
6.2. Friction 6.3. Properties of Friction 6.4. The Drag Force and Terminal Speed 6.5. Uniform Circular Motion

The Friction Force There are two types of friction forces:
Kinetic Friction Forces Static Friction Forces

Kinetic Friction Forces
An object is experiencing a kinetic friction force when the object is moving relative to a surface. Kinetic friction forces is linearly proportional to the normal force. The slope of Kinetic friction forces vs. the normal force is changing for different surfaces.

Kinetic Friction Forces
The magnitude of the kinetic friction force can be expressed as: The slope μkin is called the coefficient of kinetic friction and N is the magnitude of the normal force.

Question A person has a choice of either pushing or pulling a sled at a constant velocity, as the drawing illustrates. Friction is present. If the angle θ is the same in both cases, does it require less force to push or to pull? Account for your answer.

Example 1 Sled Riding A sled is traveling at 4.00 m/s along a horizontal stretch of snow, as Figure illustrates. The coefficient of kinetic friction is μk= How far does the sled go before stopping?

Static Friction Forces
An object is experiencing a static friction force when the object intend to move (but not move yet) relative to the surface.

Properties of Static Friction Forces
The magnitude fs of the static frictional force can have any value from zero up to a maximum value of depending on the applied force. The maximum static friction is: The μs is the coefficient of static friction, and N is the magnitude of the normal force. The coefficient of static friction is usually larger than the coefficient of kinetic friction

Example 2 The Force Needed to Start a Sled Moving
A sled is resting on a horizontal patch of snow, and the coefficient of static friction is μs= The sled and its rider have a total mass of 38.0 kg. What is the magnitude of the maximum horizontal force that can be applied to the sled before it just begins to move?

Example 3 A House on a Hill A house is built on the top of a hill with a nearby 45° slope (Fig. 6-42). An engineering study indicates that the slope angle should be reduced because the top layers of soil along the slope might slip past the lower layers. If the static coefficient of friction between two such layers is 0.5, what is the least angle φ through which the present slope should be reduced to prevent slippage?

Example 4 Block on a Slab A 40 kg slab rests on a frictionless floor. A 10 kg block rests on top of the slab (Fig. 6-58). The coefficient of static friction between the block and the slab is 0.60, whereas their kinetic friction coefficient is The 10 kg block is pulled by a horizontal force with a magnitude of 100 N. What are the resulting accelerations of (a) the block and (b) the slab?

Example 5 Body A in Fig. weighs 102 N, and body B weighs 32 N. The coefficients of friction between A and the incline are and . Angle is 40°. Let the positive direction of an x axis be up the incline. In unit-vector notation, what is the acceleration of A if A is initially (a) at rest, (b) moving up the incline, and (c) moving down the incline?

Drag Force The magnitude of the drag force is related to the relative speed: C is drag coefficient ρ is the air density (mass per volume) A is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the velocity ).

Terminal Speed When sky diving, a terminal speed will be reached when drag force is equal to the gravity.

Centripetal Force A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the body’s speed. Important: a centripetal force is not a special type of force.

Sample Problem 6 In a 1901 circus performance, Allo “Dare Devil” Diavolo introduced the stunt of riding a bicycle in a loop-the-loop (Fig. 6-10a). Assuming that the loop is a circle with radius , what is the least speed v Diavolo could have at the top of the loop to remain in contact with it there?

Sample Problem 7 Curved portions of highways are always banked (tilted) to prevent cars from sliding off the highway. When a highway is dry, the frictional force between the tires and the road surface may be enough to prevent sliding. When the highway is wet, however, the frictional force may be negligible, and banking is then essential. Figure 6-13a represents a car of mass m as it moves at a constant speed v of 20 m/s around a banked circular track of radius R=190m . (It is a normal car, rather than a race car, which means any vertical force from the passing air is negligible.) If the frictional force from the track is negligible, what bank angle θ prevents sliding?

Conceptual Questions Suppose that the coefficients of static and kinetic friction have values such that μs=2.0μk for a crate in contact with a cement floor. Does this mean that the magnitude of the static frictional force acting on the crate at rest would always be twice the magnitude of the kinetic frictional force acting on the moving crate? Give your reasoning. A box rests on the floor of an elevator. Because of static friction, a force is required to start the box sliding across the floor when the elevator is (a) stationary, (b) accelerating upward, and (c) accelerating downward. Rank the forces required in these three situations in ascending order—that is, smallest first. Explain.

During the final stages of descent, a sky diver with an open parachute approaches the ground with a constant velocity. The wind does not blow him from side to side. Is the sky diver in equilibrium and, if so, what forces are responsible for the equilibrium?

Chapter 7. Kinetic Energy and Work
7.1. What is Physics? 7.2. What Is Energy? 7.3. Kinetic Energy 7.4. Work 7.5. Work and Kinetic Energy 7.6. Work Done by the Gravitational Force 7.7. Work Done by a Spring Force 7.8. Work Done by a General Variable Force 7.9. Power

What is Physics?

Kinetic Energy Kinetic energy K is energy associated with the state of motion of an object. For an object of mass m whose speed v is well below the speed of light, Kinetic energy K is: Unit for Kinetic energy is: Kinetic energy is a scalar quantity.

Work Work W is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, Energy transferred from the object is negative work.

Finding an Expression for Work

Properties of Work Only the force component along the object’s displacement will contribute to work. The force component perpendicular to the displacement does zero work. A force does positive work when it has a vector component in the same direction displacement, A force does negative work when it has a vector component in the opposite direction. Work is a scalar quantity.

Conceptual Example The figure shows four situations in which a force acts on a box while the box slides rightward a distance across a frictionless floor. The magnitudes of the forces are identical; their orientations are as shown. Rank the situations according to the work done on the box by the force during the displacement, from most positive to most negative.

Question A shopping bag is hanging straight down from your hand as you walk across a horizontal floor at a constant velocity. Does the force that your hand exerts on the bag’s handle do any work? Explain. Does this force do any work while you are riding up an escalator at a constant velocity? Give a reason for your answer.

Example During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement while a steady wind pushes against the crate with a force The situation and coordinate axes are shown in Fig How much work does this force do on the crate during the displacement? .

Work Done by Variable Forces

Work Done by a Three-Dimensional Variable Force
The infinitesimal amount of work dW done on the particle by the force is The work W done by while the particle moves from an initial position with coordinates (x1, y1, z1) to a final position with coordinates (x2, y2, z2) is then

Net Work–Kinetic Energy Theorem
When a net external force does work Wnet on an object, the change of kinetic energy of the object equals to the net work: Where Units of work and energy are: 1 joule = 1 J =1 kg∙m2/s2 = 1 N∙m

Conceptual Example Work and Kinetic Energy
Figure illustrates a satellite moving about the earth in a circular orbit and in an elliptical orbit. The only external force that acts on the satellite is the gravitational force. For these two orbits, determine whether the kinetic energy of the satellite changes during the motion.

EXAMPLE A 2.0 kg stone moves along an x axis on a horizontal frictionless surface, acted on by only a force Fx(x) that varies with the stone's position as shown in Fig. (a) How much work is done on the stone by the force as the stone moves from its initial point at x1 = 0 to x2 = 5 m? (b) The stone starts from rest at x1 = 0 m. What is its speed at x = 8 m?

Checkpoint 1 A particle moves along an x axis. Does the kinetic energy of the particle increase, decrease, or remain the same if the particle’s velocity changes (a) from −3 m/s to −2 m/s and (b) from −2 m/s to 2 m/s? (c) In each situation, is the work done on the particle positive, negative, or zero?

EXAMPLE During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement while a steady wind pushes against the crate with a force The situation and coordinate axes are shown in Fig. How much work does this force from the wind do on the crate during the displacement? If the crate has a kinetic energy of 10 J at the beginning of displacement , what is its kinetic energy at the end of assuming ?

Example: Deep Space The space probe Deep Space 1 was launched October 24, Its mass was 474 kg. The goal of the mission was to test a new kind of engine called an ion propulsion drive, which generates only a weak thrust, but can do so for long periods of time using only small amounts of fuel. The mission has been spectacularly successful. Consider the probe traveling at an initial speed of v0=275 m/s. No forces act on it except the 56.0-mN thrust of its engine. This external force F is directed parallel to the displacement s of magnitude Determine the final speed of the probe, assuming that the mass remains nearly constant.

Example Three Forces Figure shows three forces applied to a trunk that moves leftward by 3.00 m over a frictionless floor. The force magnitudes are FA = 5.00 N, FB = 9.00 N, and FC = 3.00 N. During the displacement, (a) what is the net work done on the trunk by the three forces and (b) does the kinetic energy of the trunk increase or decrease?

Example The skateboarder in Figure a is coasting down a ramp, and there are three forces acting on her: her weight W (magnitude=675 N), a frictional force f (magnitude=125 N) that opposes her motion, and a normal force FN (magnitude=612 N). (a) Determine the net work done by the three forces when she coasts for a distance of 9.2 m. (b) If the skateboard’s initial speed is zero, what will be her final kinetic energy?

Work Done by the Gravitational Force
Work done on the ball by the gravity is: If an object is moving down, If an object is moving up, Work done by the gravity only depends on the change of height, not depends on the path.

Work Done by a Spring Force
The spring force given by Hooke’s Law: The work done by spring force:

Example In Fig., a horizontal force Fa of magnitude 20.0 N is applied to a 3.00 kg psychology book as the book slides a distance d=0.500m up a frictionless ramp at angle θ=30 degrees. (a) During the displacement, what is the net work done on the book by Fa , the gravitational force on the book, and the normal force on the book? (b) If the book has zero kinetic energy at the start of the displacement, what is its speed at the end of the displacement?

Example The only force acting on a 2.0 kg body as it moves along a positive x axis has an x component , with x in meters. The velocity at is 8.0 m/s. (a) What is the velocity of the body at ? (b) At what positive value of x will the body have a velocity of 5.0 m/s?

Power The rate at which work is done by a force is called the power.
The average power due to the work done by a force during that time interval as We define the instantaneous power P as the instantaneous rate of doing work, so that

The units of power

Sample Problem Figure 7-16 shows constant forces F1 and F2 acting on a box as the box slides rightward across a frictionless floor. Force F1 is horizontal, with magnitude 2.0 N; force F2 is angled upward by 60° to the floor and has magnitude 4.0 N. The speed v of the box at a certain instant is 3.0 m/s. What is the power due to each force acting on the box at that instant, and what is the net power? Is the net power changing at that instant?

Chapter 8. Potential Energy and Energy Conservation
8.1. What is Physics? 8.2. Work and Potential Energy 8.3. Path Independence of Conservative Forces 8.4. Determining Potential Energy Values 8.5. Conservation of Mechanical Energy 8.6. Reading a Potential Energy Curve 8.7. Work Done on a System by an External Force 8.8. Conservation of Energy

Introduction In Chapter 7 we introduced the concepts of work and kinetic energy. We then derived a net work-kinetic energy theorem to describe what happens to the kinetic energy of a single rigid object when work is done on it. In this chapter we will consider a systems composed of several objects that interact with one another.

What is Physics? (1) The system consists of an Earth–barbell system that has its arrangement changed when a weight lifter (outside of the system) pulls the barbell and the Earth apart by pulling up on the barbell with his arms and pushing down on the Earth with his feet (2) The system consists of two crates and a floor. This system is rearranged by a person (again, outside the system) who pushes the crates apart by pushing on one crate with her back and the other with her feet There is an obvious difference between these two situations. The work the weight lifter did has been stored in the new configuration of the Earth-barbell system, and the work done by the woman separating the crates seem to be lost rather than stored away.

How do we determine whether the work done by a particular type of force is “stored” or “used up.”?

The Path Independence Test for a Gravitational Force
The net work done on the skier as she travels down the ramp is given by It does not depend on the shape of the ramp but only on the vertical component of the gravitational force and the initial and final positions of her center of mass.

Path Dependence of Work Done by a Friction Force
The work done by friction along that path 1→2 is given by The work done by the friction force along path 1→4→3→2 is given by

Conservative Forces and Path Independence
conservative forces are the forces that do path independent work; Non-conservative forces are the forces that do path dependent work;

The work done by a conservative force along any closed path is zero.

Test of a System's Ability to Store Work Done by Internal Forces: the work done by a conservative internal force can be stored in the system as potential energy, and the work done by a non-conservative internal force will be “used up”

EXAMPLE 1: Cheese on a Track
Figure a shows a 2.0 kg block of slippery cheese that slides along a frictionless track from point 1 to point 2. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide?

Determining Potential Energy Values
Consider a particle-like object that is part of a system in which a conservative force acts. When that force does work W on the object, the change in the potential energy associated with the system is the negative of the work done

Gravitational Potential Energy

GRAVITATIONAL POTENTIAL ENERGY
The gravitational potential energy U is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level: SI Unit of Gravitational Potential Energy: joule (J)

Elastic Potential Energy
or we choose the reference configuration to be when the spring is at its relaxed length and the block is at .

Sample Problem 2 A 2.0 kg sloth hangs 5.0 m above the ground (Fig. 8-6). a) What is the gravitational potential energy U of the sloth–Earth system if we take the reference point y=0 to be (1) at the ground, (2) at a balcony floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y=0. (b) The sloth drops to the ground. For each choice of reference point, what is the change in the potential energy of the sloth–Earth system due to the fall?

What is mechanical energy of a system?
The mechanical energy is the sum of kinetic energy and potential energies: For example,

Conservation of Mechanical Energy
In a system where (1) no work is done on it by external forces and (2) only conservative internal forces act on the system elements, then the internal forces in the system can cause energy to be transferred between kinetic energy and potential energy, but their sum, the mechanical energy Emec of the system, cannot change. An isolated system: is a system that there is no net work is done on the system by external forces.

Example 3

Some of the following situations are consistent with the principle of conservation of mechanical energy, and some are not. Which ones are consistent with the principle? (a) An object moves uphill with an increasing speed. (b) An object moves uphill with a decreasing speed. (c) An object moves uphill with a constant speed. (d) An object moves downhill with an increasing speed. (e) An object moves downhill with a decreasing speed. (f) An object moves downhill with a constant speed.

Example 4 A Daredevil Motorcyclist
A motorcyclist is trying to leap across the canyon shown in Figure by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

EXAMPLE 5: Bungee Jumper
A 61.0 kg bungee-cord jumper is on a bridge 45.0 m above a river. The elastic bungee cord has a relaxed length of L = 25.0 m. Assume that the cord obeys Hooke's law, with a spring constant of 160 N/m. If the jumper stops before reaching the water, what is the height h of her feet above the water at her lowest point?

EXAMPLE 6 In Fig., a 2.0 kg package of tamales slides along a floor with speed v1=4.0 m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15 N, acts on it. The spring constant is 10 000 N/m. By what distance d is the spring compressed when the package stops?

Net Work on a system

Internal Work on a single rigid object

Internal Work on a system
Since Newton's Third Law tells us that the internal work is given by the integral of y the internal work on a system is not zero in general

Work-Energy Theorem

Example 7 Fireworks A 0.20-kg rocket in a fireworks display is launched from rest and follows an erratic flight path to reach the point P, as Figure shows. Point P is 29 m above the starting point. In the process, 425 J of work is done on the rocket by the nonconservative force generated by the burning propellant. Ignoring air resistance and the mass lost due to the burning propellant, find the speed vf of the rocket at the point P.

Reading a Potential Energy Curve

Finding the Force Analytically
Solving for F(x) and passing to the differential limit yield

Reading a Potential Energy Curve
Turning Points: a place where K=0 (because U=E ) and the particle changes direction. Neutral equilibrium: the place where the particle has no kinetic energy and no force acts on it, and so it must be stationary. unstable equilibrium: a point at which . If the particle is located exactly there, the force on it is also zero, and the particle remains stationary. However, if it is displaced even slightly in either direction, a nonzero force pushes it farther in the same direction, and the particle continues to move stable equilibrium: a point where a particle cannot move left or right on its own because to do so would require a negative kinetic energy

Sample Problem A 2.00 kg particle moves along an x axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy U(x) associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between x=0 and x=7m , it would have the plotted value of U. At x=6.5m , the particle has velocity v0=(-4.0m/s)i . (a) determine the particle’s speed at x1=4.5m. (b) Where is the particle’s turning point located? (c) Evaluate the force acting on the particle when it is in the region 1.9m<x<4.0m.

General Energy Conservation
For a isolated system where Wext is zero, it energy is conserved. THE PRINCIPLE OF CONSERVATION OF ENERGY: Energy can neither be created nor destroyed, but can only be converted from one form to another.

Example In Fig. 8-58, a block slides along a path that is without friction until the block reaches the section of length L=0.75m, which begins at height h=2.0m on a ramp of angle θ=30o . In that section, the coefficient of kinetic friction is The block passes through point A with a speed of 8.0 m/s. If the block can reach point B (where the friction ends), what is its speed there, and if it cannot, what is its greatest height above A?

Chapter 9. Center of Mass and Linear Momentum
9.1. What is Physics? 9.2. The Center of Mass 9.3. Newton's Second Law for a System of Particles 9.4. Linear Momentum 9.5. The Linear Momentum of a System of Particles 9.6. Collision and Impulse 9.7. Conservation of Linear Momentum 9.8. Momentum and Kinetic Energy in Collisions 9.9. Inelastic Collisions in One Dimension 9.10. Elastic Collisions in One Dimension 9.11. Collisions in Two Dimensions

central axis.) What is physics?

Defining the Position of a Complex Object
The effective “position” of the system is: The effective “position” of a system of particles is the point that moves as though all of the system’s mass were concentrated there and all external forces were applied there.

N particles system The effective position is also called as the center of mass of a system. It represents the average location for the total mass of a system

Locating a System's Center of Mass
The components of the center of mass of a system of particles are:

Velocity of center of mass

Acceleration of center of mass

EXAMPLE 1: Three Masses Three particles of masses mA = 1.2 kg, mB = 2.5 kg, and mC = 3.4 kg form an equilateral triangle of edge length a = 140 cm. Where is the center of mass of this three-particle system?

Solid Bodies If objects have uniform density,
For objects such as a golf club, the mass is distributed symmetrically and the center-of-mass point is located at the geometric center of the objects.

Question: Where would you expect the center of mass of a doughnut to be located? Why?

Checkpoint 1 The figure shows a uniform square plate from which four identical squares at the corners will be removed. (a) Where is the center of mass of the plate originally? Where is it after the removal of (b) square 1; (c) squares 1 and 2; (d) squares 1 and 3; (e) squares 1, 2, and 3; (f) all four squares? Answer in terms of quadrants, axes, or points (without calculation, of course).

EXAMPLE 2: U-Shaped Object
The U-shaped object pictured in Fig. has outside dimensions of 100 mm on each side, and each of its three sides is 20 mm wide. It was cut from a uniform sheet of plastic 6.0 mm thick. Locate the center of mass of this object.

Problem 3 Build your skill
Problem 3 Build your skill Figure 9-4a shows a uniform metal plate P of radius 2R from which a disk of radius R has been stamped out (removed) in an assembly line. Using the x-y coordinate system shown, locate the center of mass comP of the plate.

Newton's Laws for a System of Particles
is the net force of all external forces that act on the system. Msys is the total mass of the system. We assume that no mass enters or leaves the system as it moves, so that M remains constant. The system is said to be closed. is the acceleration of the center of mass of the system. Equation 9-14 gives no information about the acceleration of any other point of the system.

EXAMPLE 4: Center-of-Mass Acceleration
The three particles in Fig. a are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are FA=6 N , FB=12 N , and FC=14 N. What is the magnitude of the acceleration of the center of mass of the system, and in what direction does it move?

Collisions and Explosions
A COLLISION or EXPLOSION is an isolated event in which two or more bodies exert relatively strong forces on each other over a short time compared to the period over which their motions take place.

What is Properties of Collision?
When objects collide or a large object explodes into smaller fragments, the event can happen so rapidly that it is impossible to keep track of the interaction forces

Linear Momentum of a particle
m is the mass of the particle is its instantaneous velocity

Newton’s second law The rate of change of the momentum of a particle is proportional to the net force acting on the particle and is in the direction of that force.

The Linear Momentum of a System of Particles
M is the mass of the system

Newton's Laws The sum of all external forces acting on all the particles in the system is equal to the time rate of change of the total momentum of the system. That leaves us with the general statement:

Collision and Impulse Impulse: The average impulse <J> :
Impulse is a vector quantity It has the same direction as the force

Linear Momentum-Impulse Theorem

Suppose you are standing on the edge of a dock and jump straight down. If you land on sand your stopping time is much shorter than if you land on water. Using the impulse–momentum theorem as a guide, determine which one of the following statements is correct. a.In bringing you to a halt, the sand exerts a greater impulse on you than does the water. b.In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a greater average force. c.In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a smaller average force.

Example 1 A Well-Hit Ball
A baseball (m=0.14 kg) has an initial velocity of v0= –38 m/s as it approaches a bat. We have chosen the direction of approach as the negative direction. The bat applies an average force that is much larger than the weight of the ball, and the ball departs from the bat with a final velocity of vf=+58 m/s. (a) Determine the impulse applied to the ball by the bat. (b) Assuming that the time of contact is Δt=1.6 × 10–3 s, find the average force exerted on the ball by the bat.

Example 2 A Rain Storm During a storm, rain comes straight down with a velocity of v0=–15 m/s and hits the roof of a car perpendicularly (see Figure ). The mass of rain per second that strikes the car roof is kg/s. Assuming that the rain comes to rest upon striking the car (vf=0 m/s), find the average force exerted by the rain on the roof.

Conservation of Momentum
If no net external force acts on a system of particles, the total translational momentum of the system cannot change. Note: If the component of the net external force on a closed system is zero along an axis, then the component of the linear momentum of the system along that axis cannot change.

Conceptual Example 4 Is the Total Momentum Conserved?
Imagine two balls colliding on a billiard table that is friction-free. Use the momentum conservation principle in answering the following questions. (a) Is the total momentum of the two-ball system the same before and after the collision? (b) Answer part (a) for a system that contains only one of the two colliding balls.

Example 5 Bullet and Two Blocks In Fig. a, a 3.40 g bullet is fired horizontally at two blocks at rest on a frictionless tabletop. The bullet passes through the first block, with mass 1.20 kg, and embeds itself in the second, with mass 1.80 kg. Speeds of m/s and 1.40 m/s, respectively, are thereby given to the blocks (Fig.b). Neglecting the mass removed from the first block by the bullet, find (a) the speed of the bullet immediately after it emerges from the first block and (b) the bullet's original speed.

Example 7 The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of kg and is moving along the x axis with a velocity of +5.5 m/s. It makes a collision with puck B, which has a mass of kg and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing. Find the final speed of (a) puck A and (b) puck B.

Sample Problem 9 Two-dimensional explosion: A firecracker placed inside a coconut of mass M, initially at rest on a frictionless floor, blows the coconut into three pieces that slide across the floor. An overhead view is shown in Fig. 9-14a. Piece C, with mass 0.30M, has final speed vfc=5.0m/s. (a) What is the speed of piece B, with mass 0.20M? (b) What is the speed of piece A?

Momentum and Kinetic Energy in Collisions
If the collision occurs in a very short time or external forces can be ignored, the momentum of system is conserved. If the kinetic energy of the system is conserved, such a collision is called an elastic collision. If the kinetic energy of the system is not conserved, such a collision is called an inelastic collision. The inelastic collision of two bodies always involves a loss in the kinetic energy of the system. The greatest loss occurs if the bodies stick together, in which case the collision is called a completely inelastic collision.

Velocity of the Center of Mass
In a closed, isolated system, the velocity of the center of mass of the system cannot be changed by a collision because, with the system isolated, there is no net external force to change it.

Example of elastic collision
Two metal spheres, suspended by vertical cords, initially just touch, as shown in Fig Sphere 1, with mass m1=30 g, is pulled to the left to height h1=8.0cm, and then released from rest. After swinging down, it undergoes an elastic collision with sphere 2, whose mass m2=75 g. What is the velocity v1f of sphere 1 just after the collision?

Example of elastic collision
A small ball of mass m is aligned above a larger ball of mass M=0.63 kg (with a slight separation, as with the baseball and basketball of Fig. 9-70a), and the two are dropped simultaneously from a height of h=1.8m. (Assume the radius of each ball is negligible relative to h.) (a) If the larger ball rebounds elastically from the floor and then the small ball rebounds elastically from the larger ball, what value of m results in the larger ball stopping when it collides with the small ball? (b) What height does the small ball then reach (Fig. 9-70b)?

Example of inelastic collision
In the “before” part of Fig. 9-60, car A (mass 1100 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1400 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low μk of 0.13) stops them, at distances dA=8.2m and dB=6.1m . What are the speeds of (a) car A and (b) car B at the start of the sliding, just after the collision? (c) Assuming that linear momentum is conserved during the collision, find the speed of car B just before the collision. (d) Explain why this assumption may be invalid.

Example of completely inelastic collision
A completely inelastic collision occurs between two balls of wet putty that move directly toward each other along a vertical axis. Just before the collision, one ball, of mass 3.0 kg, is moving upward at 20 m/s and the other ball, of mass 2.0 kg, is moving downward at 12 m/s. How high do the combined two balls of putty rise above the collision point? (Neglect air drag.)

Chapter 10. Rotation 10.1. What is Physics?
10.2. The Rotational Variables 10.3. Are Angular Quantities Vectors? 10.4. Rotation with Constant Angular Acceleration 10.5. Relating the Linear and Angular Variables 10.6. Kinetic Energy of Rotation 10.7. Calculating the Rotational Inertia 10.8. Torque 10.9. Newton's Second Law for Rotation Work and Rotational Kinetic Energy

Translation and Rotation

The Rotational Variables
When an object rotates, points on the object, such as A, B, or C, move on circular paths. The centers of the circles form a line that is called the axis of rotation

In this chapter we will examine the rotations of rigid bodies about fixed rotation axes. A fixed axis means the rotation occurs about an axis that does not move

Rotational Position We choose a reference line that is perpendicular to the axis of rotation so it lies in the x-y plane. The reference line is fixed with respect to the rotating body so that it rotates around the z axis as the body rotates. We define the rotational position q of the body as the angle between the reference line at a given moment and the positive x axis, as shown in Fig.

DEFINITION OF ANGULAR DISPLACEMENT
When a rigid body rotates about a fixed axis, the angular displacement is the angle Δθ=-0 swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. SI Unit of Angular Displacement: radian (rad)

Direction of angular displacement
Right-hand rule: Curl the fingers of your right hand in the direction of the rotation. If your extended thumb points in the negative direction along the chosen axis of rotation, we call the rotational displacement negative. If your thumb points in the positive direction along the axis of rotation, the rotational displacement was positive.

DEFINITION OF AVERAGE ANGULAR VELOCITY
Instantaneous angular velocity ω:

Direction of The angular velocity
Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity.

DEFINITION OF ANGULAR ACCELERATION
Average angular acceleration: Instantaneous angular acceleration: SI Unit of Average Angular Acceleration: radian per second squared (rad/s2)

Direction of The angular acceleration
The acceleration is a vector pointing along the axis of rotation. The acceleration vector has the same direction as the change in the angular velocity: When the angular velocity is increasing, the angular acceleration vector points in the same direction as the angular velocity. When the angular velocity is decreasing, the angular acceleration vector points in the direction opposite to the angular velocity.

Example 1 A Jet Revving Its Engines
A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of –110 rad/s, where the negative sign indicates a clockwise rotation (see Figure). As the plane takes off, the angular velocity of the blades reaches –330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

Rotational and Linear Kinematics
Rotational Motion Quantity Linear Motion q Position x Δq Displacement Δx w Velocity v a Acceleration a t Time t

Rotation with Constant Rotational Acceleration
Linear Motion Rotational Motion

Example 2 Blending with a Blender
The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in, as Figure 8.11 shows. When the “blend” button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of rad (seven revolutions). The angular acceleration has a constant value of rad/s2. Find the final angular velocity of the blades.

Relating Translational and Rotational Variables
If a reference line on a rigid body rotates through an angle θ, a point within the body at a distance r from the rotation axis moves a distance s along a circular arc, where s is given by

Relating Rotational and Translational Speed
The rotational speed ω is the same at any points The translational speed is different for the points with different distance from the rotational axis.

Centripetal Acceleration and Tangential Acceleration
The nonuniform circular motion : its tangential speed is changing. An object which is in nonuniform circular motion has both centripetal acceleration ac and a tangential acceleration aT. The centripetal acceleration can be expressed in terms of the angular speed w by using vT: The magnitude of the tangential acceleration aT is: aT=ra the magnitude of the total acceleration a can be obtained by The angle f in the drawing can be determined from: Φ= tan-1 (aT/ac)

Example 3 A Helicopter Blade
A helicopter blade has an angular speed of w=6.50 rev/s and an angular acceleration of a=1.30 rev/s2. For points 1 and 2 on the blade in Figure, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations.

Example 4 A Discus Thrower
Discus throwers often warm up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. Figure (a) illustrates a top view of such a warm-up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of rad/s in a time of s before releasing it. During the acceleration, the discus moves on a circular arc of radius m. Find (a) the magnitude a of the total acceleration of the discus just before it is released and (b) the angle Φ that the total acceleration makes with the radius at this moment.

Rolling Motion (a) An automobile moves with a linear speed v.
(b) If the tires roll and do not slip, the distance d, through which an axle moves, equals the circular arc length s along the outer edge of a tire.

Kinetic Energy of Rotation
The kinetic energy of a rigid rotating body is sum over kinetic energy all the particles in the body Because v =ω r and ω is the same for all particles in a rigid rotating body, so that we have: The rotational inertia for a collection of particles is defined as:

DEFINITION OF ROTATIONAL KINETIC ENERGY
The rotational kinetic energy KER of a rigid object rotating with an angular speed ω about a fixed axis and having a rotational of inertia I is Requirement: ω must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J)

Combining Translations with Simple Rotations
An object that undergoes combined rotational and translation motion has two types of kinetic energy: a rotational kinetic energy due to its rotation about its center of mass a translational kinetic energy due to translation of its center of mass. The total mechanical energy is:

Calculating Rotational Inertia
For a continuous body, we define the rotational inertia of the body as

Which case has the largest rotational inertia?

Some Rotational Inertias

Parallel-Axis Theorem
If we know the rotational inertia of a symmetric object rotating about an axis passing through its center of mass, then the rotational inertia I about another parallel axis (where the perpendicular distance between the given axis and the axis through the center of mass is h) is:

Question Four objects having the same “radius” and mass are shown in the figure that follows. Rank the objects according to the rotational inertia about the axis shown, greatest first.

Example 5 Rolling Cylinders
A thin-walled hollow cylinder (mass=mh, radius=rh) and a solid cylinder (mass=ms, radius=rs) start from rest at the top of an incline (Figure). Both cylinders start at the same vertical height h0. All heights are measured relative to an arbitrarily chosen zero level that passes through the center of mass of a cylinder when it is at the bottom of the incline (see the drawing). Ignoring energy losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom.

Torque A net external force causes linear motion to change, but what causes rotational motion to change?

Examples of Torque

Definition of torque Magnitude:
Direction is determined by the right hand rule. Note: for fixed rotational axis motion, only components of and in the x-y plane will contribute to the torque along the rotational axis

Example 6 The Achilles Tendon
Figure (a) shows the ankle joint and the Achilles tendon attached to the heel at point P. The tendon exerts a force of magnitude F=720 N, as Figure (b) indicates. Determine the torque (magnitude and direction) of this force about the ankle joint, which is located 3.6×10–2 m away from point P.

Newton's Second Law for Rotation
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS: Requirement: a must be expressed in rad/s2. Important: are vector components along the same axis and

Example 7 The Torque of an Electric Saw Motor
The motor in an electric saw brings the circular blade from rest up to the rated angular velocity of 80.0 rev/s in rev. One type of blade has a moment of inertia of 1.41×10–2 kg · m2. What net torque (assumed constant) must the motor apply to the blade?

Example 8 Figure a shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.

Rotational Work The rotational work WR done by a constant torque in turning an object through an angle θ is: For variable torque, rotational work is: Requirement: θ must be expressed in radians. Unit of work: joule (J)

Rotational Net Work-Kinetic Energy Theorem

Power for a Rotating Body
In addition, we can find the power P associated with the rotational motion of a rigid object about a fixed axis using the equation dW = d : Note: The signs of both torque and rotational velocity are determined by the right-hand rule.

Corresponding Relations for Translational and Rotational Motion

EXAMPLE 9: Rotating Sculpture
A rigid sculpture consists of a thin hoop (of mass m and radius R = 0.15 m) and a thin radial rod (of mass m and length L = 2.0 R), arranged as shown in Fig. The sculpture can pivot around a horizontal axis in the plane of the hoop, passing through its center. (a) In terms of m and R, what is the sculpture's rotational inertia I about the rotation axis? (b) Starting from rest, the sculpture rotates around the rotation axis from the initial upright orientation of Fig. What is its rotational speed ω about the axis when it is inverted?

Sample Problem 10 A tall, cylindrical chimney will fall over when its base is ruptured. Treat the chimney as a thin rod of length L = 55.0 m (Fig a). At the instant it makes an angle of θ=35o with the vertical, what is its angular speed ωf ?

Conceptual Questions 1 The earth rotates once per day about its axis. Where on the earth’s surface should you stand in order to have the smallest possible tangential speed? Justify your answer. 2 Explain (a) how it is possible for a large force to produce only a small, or even zero, torque, and (b) how it is possible for a small force to produce a large torque.

3 A bicycle is turned upside down, the front wheel is spinning (see the drawing), and there is an angular acceleration. At the instant shown, there are six points on the wheel that have arrows associated with them. Which of the following quantities could the arrows represent: (a) tangential velocity, (b) tangential acceleration, (c) centripetal acceleration? In each case, answer why the arrows do or do not represent the quantity. 4 An object has an angular velocity. It also has an angular acceleration due to torques that are present. Therefore, the an-gular velocity is changing. What happens to the angular velocity if (a) additional torques are applied so as to make the net torque suddenly equal to zero and (b) all the torques are suddenly removed?

5 The space probe in the drawing is initially moving with a constant translational velocity and zero angular velocity. (a) When the two engines are fired, each generating a thrust of magnitude T, will the translational velocity increase, decrease, or remain the same? Why? (b) Explain what will happen to the angular velocity. 6 For purposes of computing the translational kinetic energy of a rigid body, its mass can be considered as concentrated at the center of mass. If one wishes to compute the body’s moment of inertia, can the mass be considered as concentrated at the center of mass? If not, why not?

Chapter 11. Angular Momentum
Rotational Momentum 2. Rotational Form of Newton's Second Law 3. The Rotational Momentum of a System of Particles 4. The Rotational Momentum of a Rigid Body Rotating About a Fixed Axis 5. Conservation of Rotational Momentum

Rotational Momentum A particle of mass m with translational momentum P as it passes through point A in the xy plane. The rotational momentum of this particle with respect to the origin O is: is the position vector of the particle with respect to O. Note: (1) Magnitude of rotational momentum is L=r┴ P=rP┴ (2) The particle does not have to rotate around O

Excise In the diagrams below there is an axis of rotation perpendicular to the page that intersects the page at point O. Figure (a) shows particles 1 and 2 moving around point O in opposite rotational directions, in circles with radii 2 m and 4 m. Figure (b) shows particles 3 and 4 traveling in the same direction, along straight lines at perpendicular distances of 2 m and 4 m from point O. Particle 5 moves directly away from O. All five particles have the same mass and the same constant speed. (a) Rank the particles according to the magnitudes of their rotational momentum about point O, greatest first. (b) Which particles have rotational momentum about point O that is directed into the page?

Rotational Form of Newton's Second Law
The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the rotational momentum of that particle.

Sample Problem 1 In Fig , a penguin of mass m falls from rest at point A, a horizontal distance D from the origin O of an xyz coordinate system. (The positive direction of the z axis is directly outward from the plane of the figure.). a) What is the angular momentum of the falling penguin about O? b) About the origin O, what is the torque on the penguin due to the gravitational force ?

The Rotational Momentum of a System of Particles
The total rotational momentum of a system of particles to be the vector sum of the rotational momenta of the individual particles

Newton’s Law for a System
The net (external) torque acting on a system of particles is equal to the time rate of change of the system's total rotational momentum .

The Rotational Momentum of a Rigid Body Rotating About a Fixed Axis
The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia I and its angular velocity ω with respect to that axis: Unit of Angular Momentum: kg·m2/s

CONSERVATION OF Rotational MOMENTUM
The total angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system.

examples

Corresponding Relations for Translational and Rotational Motion

Example 2 A Satellite in an Elliptical Orbit
An artificial satellite is placed into an elliptical orbit about the earth, as in Figure Telemetry data indicate that its point of closest approach (called the perigee) is rP=8.37×106 m from the center of the earth, and its point of greatest distance (called the apogee) is rA=25.1×106 m from the center of the earth. The speed of the satellite at the perigee is vP=8450 m/s. Find its speed vA at the apogee.

EXAMPLE 3 During a jump to his partner, an aerialist is to make a quadruple somersault lasting a time t = 1.87 s. For the first and last quarter revolution, he is in the extended orientation shown in Fig , with rotational inertia I1 = 19.9 kg · m2 around his center of mass (the dot). During the rest of the flight he is in a tight tuck, with rotational inertia I2 = 3.93 kg · m2. What must be his rotational speed w2 around his center of mass during the tuck?

Example 4 A uniform thin rod of length m and mass 4.00 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.00 g bullet traveling in the rotation plane is fired into one end of the rod. As viewed from above, the bullet’s path makes angle θ=60o with the rod (Fig ). If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the bullet’s speed just before impact?

Conceptual Questions 1 A woman is sitting on the spinning seat of a piano stool with her arms folded. What happens to her (a) angular velocity and (b) angular momentum when she extends her arms outward? Justify your answers. 2 A person is hanging motionless from a vertical rope over a swimming pool. She lets go of the rope and drops straight down. After letting go, is it possible for her to curl into a ball and start spinning? Justify your answer.

Chapter 15 Oscillations 15.1. What is Physics?
15.2. Simple Harmonic Motion 15.3. The Force Law for Simple Harmonic Motion 15.4. Energy in Simple Harmonic Motion 15.6. Pendulums 15.7. Simple Harmonic Motion and Uniform Circular Motion 15.9. Forced Oscillations and Resonance

What is Physics? Any measurable quantity that repeats itself at regular time intervals is defined as undergoing periodic motion. If the periodic variation of a physical quantity over time has the shape of a sine (or cosine) function, we call it a sinusoidal oscillation or simple harmonic motion. Any periodic motion is superposition of simple harmonic motions.

Simple Harmonic Motion
The maximum excursion from equilibrium is the amplitude A of the motion

The weight of an object on a vertical spring stretches the spring by an amount d 0. Simple harmonic motion of amplitude A occurs with respect to the equilibrium position of the object on the stretched spring.

Displacement

Some Observables for Simple Harmonic Motion
The period T is the time required for one complete motional cycle. The frequency f of the motion is the number of cycles of the motion per second (unit is: 1 cycle/second=1 Hz). Frequency and period are related according to: Angular frequency ω:

VELOCITY The velocity in simple harmonic motion is given by
The maximum magnitude of velocity is

ACCELERATION The acceleration in simple harmonic motion is
The maximum magnitude of the acceleration is

Force on an object in Simple Harmonic Motion
Where K=mω2 is a constant Any object under a force of will be in simple harmonic motion. This force is called restoring force.

K=mω2 is spring constant
K=mω2 is spring constant

The drawing shows plots of the displacement x versus the time t for three objects undergoing simple harmonic motion. Which object, I, II, or III, has the greatest maximum velocity?

Example 1 The diaphragm of a loudspeaker moves back and forth in simple harmonic motion to create sound, as in Figure. The frequency of the motion is f=1.0 kHz and the amplitude is A=0.20 mm. (a) What is the maximum speed of the diaphragm? (b) Where in the motion does this maximum speed occur? (c) What is the maximum acceleration of the diaphragm, and (d) where does this maximum acceleration occur?

Example 2 An 0.80-kg object is attached to one end of a spring, as in Figure, and the system is set into simple harmonic motion. The displacement x of the object as a function of time is shown in the drawing. With the aid of these data, determine (a) the amplitude A of the motion, (b) the angular frequency w, (d) the speed of the object at t=1.0 s, and (e) the magnitude of the object’s acceleration at t=1.0 s.

Energy in Simple Harmonic Motion
K=mω2 is spring constant, then m=k/ ω2

The Simple Pendulum we can write this restoring torque as
If θ is small, then sinθ~θ, For simple pendulum, I=mL2

The Physical Pendulum

Sample Problem In Fig a, a meter stick swings about a pivot point at one end, at distance h from the stick’s center of mass. (a) What is the period of oscillation T? (b) What is the distance L0 between the pivot point O of the stick and the center of oscillation of the stick?

Simple Harmonic Motion and Uniform Circular Motion
Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the circular motion occurs.

Forced Oscillations and Resonance
Two angular frequencies are associated with a system undergoing driven oscillations: the natural angular frequency ω of the system, which is the angular frequency at which it would oscillate if it were suddenly disturbed and then left to oscillate freely, and (2) the angular frequency ωd of the external driving force causing the driven oscillations. How large the displacement amplitude xm is depends on a complicated function of ωd and ω. The velocity amplitude vm of the oscillations is easier to describe: it is greatest when a condition called resonance.

Chapter 16. Wave I 16.1. What is Physics? 16.2. Types of Waves
16.3. Transverse and Longitudinal Waves 16.4. Wavelength and Frequency 16.5. The Speed of a Traveling Wave 16.7. Energy and Power of a Wave Traveling Along a String 16.9. The Principle of Superposition for Waves Interference of Waves Standing Waves Standing Waves and Resonance

What is Physics?

Types of Waves Mechanical waves. Electromagnetic waves. Matter waves.

Transverse Wave the displacement of every such oscillating string element is perpendicular to the direction of travel of the wave. This motion is said to be transverse, and the wave is said to be a transverse wave.

Longitudinal Waves The motion of the elements of air is paralle to the direction of the wave’s travel, the motion is said to be longitudinal, and the wave is said to be a longitudinal wave.

Description of a wave

Wavelength and Angular Wave Number
The wavelength of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions of the shape of the wave (or wave shape) the angular wave number of the wave

Period, Angular Frequency, and Frequency
the period of oscillation T of a wave to be the time any string element takes to move through one full oscillation. Angular frequency is The frequency f of a wave is defined as 1/T and is related to the angular frequency by

The Speed of a Traveling Wave
the wave speed as

Checkpoint Here are the equations of three waves. Rank the waves according to their (a) wave speed and (b) maximum speed perpendicular to the wave’s direction of travel (the transverse speed), greatest first. (1) (2) (3)

Sample Problem A wave traveling along a string is described by
in which the numerical constants are in SI units ( m,72.1 rad/m, and 2.72 rad/s). What is the amplitude of this wave? What are the wavelength, period, and frequency of this wave?

Wave Speed on a Stretched String
The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.

Energy and Power of a Wave Traveling Along a String
Elastic Potential Energy: When the string element is at its (a) position (element a in its length has its normal undisturbed value dx, so its elastic potential energy is zero. However, when the element is rushing through its (b) position, it has maximum stretch and thus maximum elastic potential energy. Kinetic Energy : A string element of mass dm, oscillating transversely in simple harmonic motion as the wave passes through it, has kinetic energy associated with its transverse velocity. It is maximum at position (b) and is zero at position (a).

Energy Transport The wave transports the energy along the string.

The Rate of Energy Transmission

Sample Problem A string has linear density and tension We send a sinusoidal wave with frequency Hz and amplitude along the string. At what average rate does the wave transport energy?

The Principle of Superposition for Waves
Overlapping waves algebraically add to produce a resultant wave (or net wave). Overlapping waves do not in any way alter the travel of each other.

Standing Waves If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave. Applying the trigonometric relation of Eq leads to Applying the trigonometric relation of Eq leads to

Feature of Standing Wave
The nodes, are places along the string where the string never moves. antinodes are halfway between adjacent nodes where the amplitude of the resultant wave is a maximum.

Standing Waves In a traveling sinusoidal wave, the amplitude of the wave is the same for all string elements. That is not true for a standing wave, in which the amplitude varies with position. Applying the trigonometric relation of Eq leads to Applying the trigonometric relation of Eq leads to

Checkpoint 5 Two waves with the same amplitude and wavelength interfere in three different situations to produce resultant waves with the following equations. In which situation are the two combining waves traveling (a) toward positive x, (b) toward negative x, and (c) in opposite directions?

Standing Waves and Resonance
For certain frequencies, the interference produces a standing wave pattern (or oscillation mode) Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies.

Example In Fig , a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L=1.2m, linear density μ=1.6g/m, and the oscillator frequency f=120Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m=1.0kg? ,

Chapter 1. Measurement 1.What is Physics? 2. Measuring Things

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Chapter 1. Measurement 1.What is Physics? 2. Measuring Things

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