Presentation on theme: "Review: using F=ma Force in circular motion"— Presentation transcript:
1 Review: using F=ma Force in circular motion 𝑎 𝑐 and 𝑎 𝑡 in curvilinear motion (e.g. non-uniform circular motion)Note on other important forcesPropulsionResistive force (Drag in air/liquid), terminal velocicty 𝑣 𝑇Conservative force, spring force Potential energy
2 Relevant concepts:“inertia” and forcea) Inertial reference frameb) accelerated frame of the car- fictitious force
5 Conical Pendulum (e.g. ชิงช้าหมุน) The object is in equilibrium in the vertical direction.It undergoes uniform circular motion in the horizontal direction.∑𝐹𝑦 = 0 → 𝑇 cos 𝜃 = 𝑚𝑔∑𝐹𝑥 = 𝑇 sin𝜃 = 𝑚 𝑎𝑐v is independent of m
6 Car traveling in a banked curve Design the curve with no frictionin equilibrium in the vertical direction.in uniform circular motion in the horizontal direction a component of the normal force supplies the centripetal force.The angle of bank isNote:The banking angle is independent of the mass of the vehicle.If the car rounds the curve at less than the design speed, friction is necessary to keep it from sliding down the bank.If the car rounds the curve at more than the design speed, friction is necessary to keep it from sliding up the bank.
7 Car traveling in a horizontal (Flat) Curve uniform circular motion in the horizontal direction.in equilibrium in the vertical direction.The force of static friction supplies the centripetal force.The maximum speed at which the car can negotiate the curve is:Note: this does not depend on the mass of the car.Section 6.1
8 Ferris Wheel 𝑭 𝒏𝒆𝒕 𝑭 𝒏𝒆𝒕 = 𝑚 𝑣 2 𝑅 (− 𝒓 ) 𝑭 𝒏𝒆𝒕 Uniform circular motion with constant speed v (controlled by the motor)Under gravity, the child feels apparent weight differently at top and bottom𝑭 𝒏𝒆𝒕𝑭 𝒏𝒆𝒕 = 𝑚 𝑣 2 𝑅 (− 𝒓 )𝑭 𝒏𝒆𝒕Section 6.1
9 Ferris Wheel (2)At the bottom of the loop, the upward force (the normal) experienced by the object is greater than its weight.At the top of the circle, the force exerted on the object is less than its weight.Section 6.1
10 Non-uniform Circular Motion If the speed also changes in magnitudethere is non-zero tangential accelerationSection 6.2
11 Vertical Circle with Non-Uniform Speed Launch by shooting (say, from the bottom at initial velocity 𝒗 𝒃𝒐𝒕 ). The string restricts motion to a circle of radius R (1 DOF)What is T and acceleration?Tension depends on position 𝜃(𝑡) and speed 𝑣 𝑡 = d𝜃 𝑑𝑡 𝑅 ,both varying with timeThe tension at the bottom is a maximum.The tension at the top is a minimum.Requires critical speed 𝑣 𝑡𝑜𝑝 ≥ 𝑅𝑔 to complete the circle ( 𝑇 𝑡𝑜𝑝 ≥ 0 at 𝜃 =𝜋)Section 6.2
12 Resistive force (or drag force in air/fluid) and terminal velocity 𝑣 𝑇 (this topic is not in the test)
13 1. Linear (Stokes law in liquid) 𝑭 𝑫 ∝− 𝒗 𝐹 𝐷 = −𝑏𝑣 Resistive force 𝑭 𝑫 could be of either form(linear or quadratic to 𝑣), but always opposite direction to motion (− 𝒗 )1. Linear (Stokes law in liquid)𝑭 𝑫 ∝− 𝒗𝐹 𝐷 = −𝑏𝑣2. Quadratic (air resistance)𝑭 𝑫 ∝− 𝑣 2𝐹 𝐷 = − 1 2 𝐶 𝐷 𝜌𝐴 𝑣 2𝐶 𝐷 is the drag coefficient (empirical, no dimension, typically 𝐶 𝐷 ≈ 1 2 ).𝜌 is the density of air.𝐴 is the cross-sectional area of the object𝑣 𝑇 ≈ 4𝑚𝑔 𝜌𝐴
16 Outline Work done by a constant force Projection and scalar product of vectorsForce that results in positive work.negative work? forces that do no work?Work done by varying force with displacementWork-energy theorem
19 Work, cont.W = F Dr cos qA force does no work on the object if the force does not move through a displacement.The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application.Section 7.2
23 6.1 Work Done by a Constant Force Example 3 Accelerating a CrateThe truck is accelerating ata rate of m/s2. The massof the crate is 120-kg and itdoes not slip. The magnitude ofthe displacement is 65 m.What is the total work done onthe crate by all of the forcesacting on it?
24 6.1 Work Done by a Constant Force The angle between the displacementand the friction force is 0 degrees.
26 Scalar Product, cont The scalar product is commutative. The scalar product obeys the distributive law of multiplication.Section 7.3
27 Dot Products of Unit Vectors Using component form with vectors: In the special case whereSection 7.3
28 6.2 The Work-Energy Theorem and Kinetic Energy Consider a constant net external force acting on an object.The object is displaced a distance s, in the same direction asthe net force.The work is simply
29 6.2 The Work-Energy Theorem and Kinetic Energy DEFINITION OF KINETIC ENERGYThe kinetic energy KE of and object with mass mand speed v is given by
30 6.2 The Work-Energy Theorem and Kinetic Energy When a net external force does work on and object, the kineticenergy of the object changes according to
31 6.2 The Work-Energy Theorem and Kinetic Energy Example 4 Deep Space 1The mass of the space probe is 474-kg and its initial velocityis 275 m/s. If the 56.0-mN force acts on the probe through adisplacement of 2.42×109m, what is its final speed?
34 6.2 The Work-Energy Theorem and Kinetic Energy In this case the net force is
35 6.2 The Work-Energy Theorem and Kinetic Energy Conceptual Example 6 Work and Kinetic EnergyA satellite is moving about the earthin a circular orbit and an elliptical orbit.For these two orbits, determine whetherthe kinetic energy of the satellitechanges during the motion.W<0W>0
37 6.3 Gravitational Potential Energy Work due to gravity is independent of path!We call this phenomenon “conservative force” We can define “potential energy” based on this force
38 6.3 Gravitational Potential Energy Example 7 A Gymnast on a TrampolineThe gymnast leaves the trampoline at an initial height of 1.20 mand reaches a maximum height of 4.80 m before falling backdown. What was the initial speed of the gymnast?
39 6.3 Gravitational Potential Energy DEFINITION OF GRAVITATIONAL POTENTIAL ENERGYThe gravitational potential energy PE is the energy that anobject of mass m has by virtue of its position relative to thesurface of the earth. That position is measured by the heighth of the object relative to an arbitrary zero level:
40 6.4 Conservative Versus Nonconservative Forces DEFINITION OF A CONSERVATIVE FORCEVersion 1 A force is conservative when the work it doeson a moving object is independent of the path between theobject’s initial and final positions.Version 2 A force is conservative when it does no workon an object moving around a closed path, starting andfinishing at the same point.
45 6.4 Conservative Versus Nonconservative Forces An example of a nonconservative force is the kineticfrictional force.The work done by the kinetic frictional force is always negative.Thus, it is impossible for the work it does on an object thatmoves around a closed path to be zero.The concept of potential energy is not defined for anonconservative force.
46 6.4 Conservative Versus Nonconservative Forces In normal situations both conservative and nonconservativeforces act simultaneously on an object, so the work done bythe net external force can be written as
47 Spring force and Spring (Elastic) Potential Energy
48 6.4 Conservative Versus Nonconservative Forces Work done by conservative forceWork done by non-conservative forceTHE WORK-ENERGY THEOREM
49 6.5 The Conservation of Mechanical Energy If the net work on an object by nonconservative forcesis zero, then its energy does not change:
50 6.5 The Conservation of Mechanical Energy THE PRINCIPLE OF CONSERVATION OFMECHANICAL ENERGYThe total mechanical energy (E = KE + PE) of an objectremains constant as the object moves, provided that the network done by external nononservative forces is zero.
52 6.5 The Conservation of Mechanical Energy Example 8 A Daredevil MotorcyclistA motorcyclist is trying to leap across the canyon by drivinghorizontally off a cliff 38.0 m/s. Ignoring air resistance, findthe speed with which the cycle strikes the ground on the otherside.
55 6.5 The Conservation of Mechanical Energy Conceptual Example 9 The Favorite Swimming HoleThe person starts from rest, with the ropeheld in the horizontal position,swings downward, and then letsgo of the rope. Three forcesact on him: his weight, thetension in the rope, and theforce of air resistance.Can the principle ofconservation of energybe used to calculate hisfinal speed?
56 DEFINITION OF AVERAGE POWER Average power is the rate at which work is done, and itis obtained by dividing the work by the time required toperform the work.
59 6.8 Other Forms of Energy and the Conservation of Energy THE PRINCIPLE OF CONSERVATION OF ENERGYEnergy can neither be created not destroyed, but canonly be converted from one form to another.
60 Thermal energy (heat) as internal energy of materials Where does work due to nonconservative force (Wnc) go to, other than kinetic energy?Thermal energy (heat) as internal energy of materialsThis means energy of the universe (isolated system) is always conserved.When we say “not energy conserving” we mean just in the sub-system.
63 7.1 The Impulse-Momentum Theorem There are many situations when theforce on an object is not constant.
64 7.1 The Impulse-Momentum Theorem DEFINITION OF IMPULSEThe impulse of a force is the product of the averageforce and the time interval during which the force acts:Impulse is a vector quantity and has the same directionas the average force.
68 7.1 The Impulse-Momentum Theorem DEFINITION OF LINEAR MOMENTUMThe linear momentum of an object is the productof the object’s mass times its velocity:Linear momentum is a vector quantity and has the samedirection as the velocity.
70 7.1 The Impulse-Momentum Theorem When a net force acts on an object, the impulse ofthis force is equal to the change in the momentumof the objectimpulsefinal momentuminitial momentum
71 7.1 The Impulse-Momentum Theorem Example 2 A Rain StormRain comes down with a velocity of -15 m/s and hits theroof of a car. The mass of rain per second that strikesthe roof of the car is kg/s. Assuming that rain comesto rest upon striking the car, find the average forceexerted by the rain on the roof.
72 7.1 The Impulse-Momentum Theorem Neglecting the weight ofthe raindrops, the net forceon a raindrop is simply theforce on the raindrop due tothe roof.
73 7.1 The Impulse-Momentum Theorem Conceptual Example 3 Hailstones Versus RaindropsInstead of rain, suppose hail is falling. Unlike rain, hail usuallybounces off the roof of the car.If hail fell instead of rain, would the force be smaller than,equal to, or greater than that calculated in Example 2?
75 7.2 The Principle of Conservation of Linear Momentum If the sum of the external forces is zero, thenPRINCIPLE OF CONSERVATION OF LINEAR MOMENTUMThe total linear momentum of an isolated system is constant(conserved). An isolated system is one for which the sum ofthe average external forces acting on the system is zero.
76 7.2 The Principle of Conservation of Linear Momentum Conceptual Example 4 Is the Total Momentum Conserved?Imagine two balls colliding on a billiardtable that is friction-free. Use the momentumconservation principle in answering thefollowing questions. (a) Is the total momentumof the two-ball system the same beforeand after the collision? (b) Answerpart (a) for a system that contains onlyone of the two collidingballs.
77 7.2 The Principle of Conservation of Linear Momentum Example 6 Ice SkatersStarting from rest, two skaterspush off against each other onice where friction is negligible.One is a 54-kg woman andone is a 88-kg man. The womanmoves away with a speed of+2.5 m/s. Find the recoil velocityof the man.
78 7.2 The Principle of Conservation of Linear Momentum
79 7.2 The Principle of Conservation of Linear Momentum Applying the Principle of Conservation of Linear Momentum1. Decide which objects are included in the system.2. Relative to the system, identify the internal and external forces.3. Verify that the system is isolated.4. Set the final momentum of the system equal to its initial momentum.Remember that momentum is a vector.
81 7.3 Collisions in One Dimension The total linear momentum is conserved when two objectscollide, provided they constitute an isolated system.Elastic collision -- One in which the total kineticenergy of the system after the collision is equal tothe total kinetic energy before the collision.Inelastic collision -- One in which the total kineticenergy of the system after the collision is not equalto the total kinetic energy before the collision; if theobjects stick together after colliding, the collision issaid to be completely inelastic.
83 7.3 Collisions in One Dimension Example 8 A Ballistic PendulumThe mass of the block of woodis 2.50-kg and the mass of thebullet is kg. The blockswings to a maximum height of0.650 m above the initial position.Find the initial speed of thebullet.
84 7.3 Collisions in One Dimension Apply conservation of momentumto the collision:Inelastic collision
85 7.3 Collisions in One Dimension Applying conservation of energyto the swinging motion:
89 In an isolated system, the total linear momentum does not change, 7.5 Center of MassIn an isolated system, the total linear momentum does not change,therefore the velocity of the center of mass does not change.