Instructor: Laura Fellman PHY 101 Lecture Notes Instructor: Laura Fellman

Chapter 2 A brief look at the historical development of physics and Newton’s 1st Law of Motion

Aristotle (384-322 BC) Greek philosopher/scientist Aristotle was an observer not an experimenter He thought there were 2 classes of motion: (1) natural motion: every object in universe has a proper place and strives to get to this place (2) violent motion = imposed motion results from pushing and pulling forces WE NOW KNOW ARISTOTLE WAS WRONG!

Nicolaus Copernicus (1473-1543) Polish astronomer who changed astronomy profoundly 1510: derived a heliocentric or “sun-centered” model Only published in 1543: “De Revolutionibus” Book was banned by Church between 1610 & 1835 Now we recognize Copernicus as a “giant” in astronomy

Galileo Galilei (1561 – 1642) professor of Mathematics at University in Italy Galileo used observations and experiments to disprove Aristotle’s ideas he was interested in HOW things moved, not why they moved. we call this kinematics Important experiment: Galileo dropped heavy and light objects together and found they hit the ground at the same time. See the experiment in action

Air resistance Air resistance affects motion and makes it more complicated See Elephants and feathers If we can ignore air resistance, we find that the relationships describing motion are simpler When can we neglect air resistance? (1) If there is no air! (in a vacuum) (2) If the objects in motion are: heavy compact (dense) traveling at moderate speeds

Back to Galileo Galileo stated that: If there is no interference with a moving object, it will keep moving in a straight line forever. See Web demo example Consider an experiment in which you: roll a ball up an incline roll a ball down an incline along a flat surface see Figure 2.3 in text and an online explanation

Galileo & the telescope In 1608 a Dutch lens maker invented the telescope Galileo built one in 1609 In 1610 he published “The Starry Messenger” documenting many important observations, including Moon’s surface had features (mountains & valleys) Milky Way was made up of many stars Jupiter had moons circling it Soon after this he also discovered: Sun was not perfect but had “spots” on its surface Sun was spherical & rotated about its own axis Venus went through complete set of phases like Moon

Galileo in trouble In 1632 Galileo publishes “Dialogue Concerning the Two Chief World Systems” defending Copernicus Interrogated by the Inquisition In 1633 he recants and admits his errors Sentenced to life house arrest where he dies In 1992 Catholic church finally officially admits that Galileo was right

Newton(1642-1727) Changed the focus from “how” to “why” Made brilliant contributions to physics! Pondered why apple fell to Earth amongst other things He summarized his findings in 3 laws = Newton’s Laws All involve the idea of a force (or lack of a force)

Isaac Newton: Yes, the apple really fell! Published “Principia” in which he outlined 3 basic laws of motion: A body continues at rest or in motion in a straight line unless acted on by some force. The change in motion of a body is proportional to the size and direction of the force acting on it. When one body exerts a force on a 2nd body, the 2nd body exerts an equal & oppositely directed force on the first.

Newton’s First Law/ Law of Inertia An object at rest remains at rest if no force acts on it An object in motion remains in motion if no force acts on it Inertia = resistance of an object to a change in its motion See this in action Experience tells us that the heavier an object is, the harder it is to get it up to speed when pushing it. Scientifically we could say: the greater the object’s mass, the greater its resistance to a change in its motion. So mass is a measure of an object’s inertia.

Force Can think of force as a push or pull action What causes this push or pull? Contact force Electrical force Magnetic force non-contact force Gravitational force Forces result in a change of motion What if more than one force acts at a time?

Net force Fnet ? Fnet ? Fnet ? Need to combine the forces & find net force 3N 2N 4N Fnet ? Fnet ? Fnet ?

Review of Law of Inertia See this online summary

Equilibrium Condition for equilibrium: Fnet = 0 Static equilibrium: so all forces balance each other Static equilibrium: speed = 0 (no motion), and Fnet = 0 Support forces Q. What stops a book from falling through the table it lies on? Ans: A support or “Normal” force What’s normal about it?

Examples: How does a scale work? Standing on one scale: Identify what forces are involved what is the sum of these forces? Spring stretches (compresses) by an amount proportional to force that pulls (pushes) on it See this in action Standing on one scale: What is the net force? Now stand on 2 scales: what does each scale read? How would scale readings change if you shift your weight?

Tension Tension (T) is a type of force (like gravitational force or electric force are force types) It is a “pulling” force usually exerted on an object by a rope or a chain Pulleys: change direction of force, not the magnitude 1 2 3 T1 , T2 and T3 are all equal in size, but in different directions.

Examples: Window washers: Joe and Jane (equal weights) What are T1 and T2 ? What if Jane, on right, walks over towards Joe? What happens to T1 and T2 now ? What happens to the total tension (T1 + T2 ) How are T1 and T2 related to each other? spring scale T1 = ? T2 = ? spring scale T1 = ? T2 = ?

Let’s try practice pages 3 and 4 now in your Practicing Physics book Then we’ll try this question……

Dynamic equilibrium Conditions for “moving” equilibrium: Example: Still need net force on object = 0 object moves at constant velocity Example: Flying at constant speed in airplane Key is you can’t feel that you are moving When do we get a sensation of motion?

Chap 3: Linear Motion Let’s find ways to describe how things move

Description of Motion We will consider motion in terms of: distance, and time Graphs are a great way to visualize motion. First consider only position or distance from a point: 0 1 2 3 4 x-axis in meters object starts at zero marker and moves, in 1 meter steps, to the 3 meter mark

As before we can graph our position but now in relation to time Now we include time record where the object is and when it gets there As before we can graph our position but now in relation to time position (x) in [m] 0 1 2 3 4 time (t) in [seconds, s] See motion being graphed in passing lane demo 4 3 2 1

Distance and time We can combine distance and time knowledge to get the following quantities: Speed: how fast? Velocity (v): how fast and in what direction? Acceleration (a): how quickly does v change?

Speed: how fast? distance speed = time Units: km/hour or mph or m/s Two ways to look at speed: (1) average speed (2) instantaneous speed SI Unit for speed

Average speed Objects don’t always travel at same speed Example: driving your car drive to Seattle (180 miles) in 3 hours may stop, get stuck in traffic, etc Can still determine my average speed: total distance covered average speed = time interval

Instantaneous speed Speed at any one instant Example: when driving your speed changes instantaneous speed = speed on your speedometer Special case: if your speed is constant for whole journey, then: instantaneous speed at all times = average speed

Graphing speed vs time position (x) Just like we graphed position vs time, we can graph velocity as it changes with time. position (x) in [m] 0 1 2 3 4 time (t) 0 1 2 3 4 time (t) Let’s go back to the passing lane demo and graph v vs time now instead of x vs time. 4 3 2 1 4 3 2 1 velocity (v) in [m/s]

Examples involving distance and speed Let’s try some conceptual questions: Motorist Bikes and Bees

More on average speed A reconnaissance plane flies 600 km away from its base at 200 km/h, then it flies back to its base at 300 km/h. What is the plane’s average speed?

Velocity Now we consider speed and direction Example: speed = 50 km/h velocity = 50 km/h to the south constant speed: equal distances covered in equal time intervals constant velocity = constant speed and no change in direction Ex 1: car moves around a circular track constant speed but velocity not constant!

Speed vs Velocity Here is example where average speed and average velocity are very different. Example: Walking the dog The owner and the dog have the same change in position but the dog covers much more distance in the same time, so they have the same average velocity but very different average speeds. See also a similar online demo of this idea

Acceleration Acceleration = rate of change of velocity time interval acceleration: speeding up or slowing down Q. Can we feel velocity? Q. Can we feel acceleration? Q. What controls in a car make it accelerate?

Examples Ex 1: A car starts at rest and reaches 60 mi/hr in 10s. Q. What is the car’s acceleration? Acceleration = (change in v) = 60 mi/hr = 6 mi/hr.s time 10 s Ex 2: A cyclist’s speed increases from 4 m/s to 10 m/s in 3 seconds. Q. What is the cyclist’s acceleration?

Graphs showing acceleration What does a velocity vs time graph look like when an object is accelerating? Let’s go back to our car demo and see what this looks like in the stoplight scenario Now lets look at 3 graphs of the same motion: position vs time velocity vs time acceleration vs time

Acceleration on inclined planes Q. On which of these hills does the ball roll down with increasing speed and decreasing acceleration along the path? A B C (Hint: see Fig 3.6 in textbook)

(meters per second squared) Free fall Things fall due to the force of gravity if there are no restraints (air resistance) on object, we say the object is in FREE FALL acceleration due to gravity is approximately g = 10 m / s2 (meters per second squared) The actual value is closer to g = 9.8 m / s2 When objects fall, we will ask…….. How fast? How far?

How fast and how far? Q. If an object is dropped from rest (no initial velocity) at the top of a cliff, how fast will it be travelling: after 1 second? after 2 seconds? Q. How far does object drop in 1s? Why?

Summary: Motion relationships Instantaneous velocity for an object that starts at rest: v = acceleration * time (in general) = gravity * time (for free fall object) or for an object that starts with an initial speed v = initial velocity + a * t = initial velocity – g * t (up is positive) Distance traveled for an object that starts at rest: d = ½ acceleration * (time)2 (general) = ½ g * t 2 (for free fall) Distance traveled for an object that starts with an initial speed d = initial velocity * time + ½ acceleration * (time)2 = initial velocity * t - 1/2 g t2 Remember to use correct units: if g has units of m / s2 then you must use time in seconds.

Examples Look over Practice pages 5 and 6 Example: A ball is dropped from rest from a height of 20m. How long does it take to reach the ground?

Chapter 4: Newton’s 2nd Law Why things move

Newton’s 2nd Law of Motion The acceleration (a) of an object is: directly proportional to the net force (Fnet) acting on it, and inversely proportional to the mass (m) of the object In symbols we can write: a = Fnet / m NOTE: acceleration and force both have a direction and a magnitude associated with them direction of “a” is given by the direction of Fnet

Notation: N F W Normal force (contact force) Pulling or pushing force Weight (gravitational force) W N Normal force (contact force) F Pulling or pushing force Example: If the block has a mass of 10 kg and if pulled by a force of 50N, find the values of the forces shown in the above diagram and calculate the horizontal acceleration.

Rank the accelerations, smallest to largest A B C D

Mass, Weight & Volume Mass: how much “stuff” something is made of measure of an object’s inertia: more mass = more inertia UNITS of measurement: [kg] or [grams] Weight: force on an object due to gravity UNITS of measurement: [Newton, N] (metric unit) or [pounds, lbs] Volume: mass is not volume! Massive doesn’t mean voluminous something can be massive (heavy) but not large this object has a high density = (mass) / (volume)

Examples What are the mass and weight of a 10 kg block on: (a) the Earth (b) moon A 50 kg woman in an elevator is accelerating upward at a rate of 1.2 m/s2. (a) What is the net force acting on the woman? (b) What is the gravitational force acting on her? (c) What is the normal force pushing upward on the woman’s feet? See a demo of an elevator ride in action

Newton’s 2nd Law in many object problems Let’s try an example where there are several objects involved: Three blocks of equal mass (2kg) are tied together. If you pull on one end with a force of 30N, what are the tensions in the other two ropes that join the blocks together? 2kg T1 = 30N T2 = ? T3 = ?

Now we have Dynamic Equilibrium Conditions: v = constant & a = 0 Friction Now we are ready to start considering the effects of friction drag a block across surface know there is friction between surface and block if speed of the block, v = constant, then a = 0 so by Newton’s 2nd Law: Fnet = 0 Now we have Dynamic Equilibrium Conditions: v = constant & a = 0

Friction Need: 2 surfaces are in mutual contact magnitude of frictional force? depends on the type of surfaces in contact Which is harder to push? depends on the weight of the object direction of frictional force? in opposite direction to motion What causes the friction? Irregularities (roughness) in surfaces Direction of motion Frictional force

2 kinds of friction Static friction > sliding friction Static friction: before there is any motion Sliding friction: when block is motion Static friction > sliding friction Friction = 70N Applied force = 70N v = 0 Friction = 50N Applied force = 50N v = constant

Interesting facts about friction Does not depend on: speed and contact area So then: Why do trucks have so many tires? Why do high performance cars have wide tires?

Force at angles & friction support force tension vertical part of tension friction horizontal part of tension weight

Free-fall revisited Let’s ignore air drag just for a moment: BUT A heavy object experiences a larger gravitational force so you might think that it has a larger acceleration than a light object (a ~ F) BUT The heavy object has greater inertia & so it has a greater resistance to change in it motion so you might think it has a smaller acceleration than a light object (a ~ 1/m) Actually: combine both of these and get that the two objects have the same acceleration! a ~ Fnet / m (see Fig 4.10)

Friction in fluids (drag) Fluids = things that flow gas (e.g. air) or liquid (e.g. water) What does drag, or resistance in fluids, depend on? properties of fluid (density) speed (in lab we will determine the exact relationship) area of contact So friction in fluids is very different to friction between 2 solids in contact!

Non-Free fall 2 equal masses are dropped, but have different surface areas. Which hits the ground first? 2 parachuters (one heavy, one light) jump out of an airplane. Which one of the two falls faster? What is going on? free fall: only gravitational force (weight) so net force Fnet = W non-free fall: Must now also consider the air resistance (R) now net force Fnet = W - R

Terminal velocity Terminal velocity is achieved when falling object is no longer accelerating (a = 0, v = max) since Fnet = W - R Acceleration: a = Fnet / m = (W - R) / m so a = 0 when W = R Recall that R depends on speed, so as speed increases R increases until eventually R = W If W is small then R = W sooner (at a lower velocity) then for large W So if 2 objects are the same size, the heavier one will have a greater terminal velocity

Let’s consider some examples First let’s look at the force of air resistance Let’s revisit the elephant and the feather falling Now let’s see what happens during a skydiver’s journey to the ground First you try practice page 10 Then we’ll look at a demo to see the jump and forces in action: Animated skydiver

Chapter 5: Newton’s 3rd Law and Vectors

Newton’s 3rd Law Whenever object A exerts a force on object B, object B exerts an equal and opposite force on object A. refer to these as action & reaction forces see Fig 5.5 Hand pushes on table (action) Table pushes on hand (reaction) How to identify these force “pairs”: always involves 2 forces the forces are acting on different objects

Examples and Problems First we’ll consider the question of an apple on a table. Now look at these examples of force pairs And try the tutorial at your textbook website Example: A horse pulls a cart. If the cart exerts a force on the horse that is equal and opposite to the force that the horse exerts on the cart, why does the cart move? Again see the textbook website tutorial for more

Defining your system So do action/reaction forces cancel each other? No! Careful: they are not acting on the same body! Example: apple pulls on orange in a cart Consider 3 different systems: 1. The orange only 2. The apple only 3. Orange & apple together Frictional force = external force

Combining Newton’s 2nd and 3rd laws Example 1: Find: (a) acceleration of the blocks (b) force on block B by block A (c) force on block A by block B 5 kg 10 kg A B 150 N Ice, so no friction to worry about

Example 2: A 56 kg parent and a 14 kg child are ice skating Example 2: A 56 kg parent and a 14 kg child are ice skating. They face each other and push on each other’s hands. (a) Which person experiences a bigger force? (b) Is the acceleration of the child larger, the same, or smaller than the parent’s acceleration? (c) If the acceleration of the child is 2.6 m/s2, what is the parent’s acceleration?

2D motion and Vectors

2-D Motion Till now: 1-D motion: motion along a line position, speed, velocity and acceleration Now: 2-D motion Motion in the horizontal & vertical directions or in a circle will need a new way to represent this motion Several topics related to 2-D motion circular motion (return to this later) relative motion covered at end of Chap 5 vectors projectile motion (beginning of Chap 10)

Relative Motion Airplane flies: same is true when you ride your bike! tailwind tailwind headwind Airplane flies: faster with a tailwind slower into a headwind same is true when you ride your bike! What happens in a crosswind? now have 2-D motion Need to introduce vectors headwind

Vectors Imagine the following: Vectors: arrows that illustrate both: you’re riding on the bus with a physicist you decide to ask her how things work all the physicist has on hand is an envelope What happens? Vectors: arrows that illustrate both: size Direction Examples: Scalars: only size

Vector example (1-D) Consider the airplane: wind Consider the airplane: airplane’s velocity: vA = 100 km/h to north tailwind: vw = 20 km/h to north What is the plane’s speed relative to the ground? vR = vA + vw = 100 km/h + 20 km/h = 120 km/h Now consider a headwind: vw = 20 km/h to south vR = vA + vw = 100 km/h + (-20 km/h) = 80 km/h plane resultant wind plane resultant

Crosswind (2-D) airplane’s velocity: vA = 80 km/h to north crosswind: vw = 60 km/h to east Want to add 2 vectors and get the resultant vector Use parallelogram method: complete “box” by adding parallel lines draw a diagonal from the starting point of 2 vectors To find the length of the diagonal (resultant): scale drawing (measure) Pythagorean Theorem: c2 = a2 + b2 or c = a2 + b2 To find the direction of the resultant: scale drawing (measure with a protractor) use trigonometry

Chapter 10: Projectile Motion a. k. a Chapter 10: Projectile Motion a.k.a. How to hit your neighbors with a cannon ball! ?

Projectile motion When an object is given: an initial horizontal velocity experiences the force of gravity (vertical direction) we call the object a projectile the path the object follows is its trajectory How do we determine the projectiles trajectory? We note that: THE VERTICAL AND HORIZONTAL MOTION OF AN OBJECT DO NOT AFFECT EACH OTHER! Dropping ball demo

Target practice: Horizontal Launch Remember: we can consider the horizontal component of motion and the vertical component separately! If I aim directly at the target and it takes my arrow 1 second to reach the target, where does my arrow end up? Horizontal projectile launch ? If you say it hits below, how far below the target?

Now add in the velocity components

Firing at an angle How do we figure out its trajectory? Let’s consider what happens when a Zookeeper fires a banana at a monkey And then let’s see this in action Now consider Fig 10.6 in your textbook The cannon now fires upward at some angle How do we figure out its trajectory? First, consider what path projectile will follow if gravity was not present (in other words, a straight line) Then after each second, consider how far projectile would fall straight down We see that the trajectory of the projectile has the mathematical shape known as a parabola

Velocity of a projectile Let’s take a look at a demo of cannonball that is fired at an angle. The horizontal and vertical parts (components) of the ball’s velocity are shown We can combine these two components by adding them as vectors using the parallelogram method to give us the velocity of the ball at any point. Important: no acceleration in horizontal direction so projectile moves equal horizontal distances in equal time intervals.

How high & how far? What do we know when we fire the cannonball? its launch angle its launch (muzzle) speed Then we might want to know: How high does it go? = vertical part of the problem How far does it go? = horizontal part of the problem KEY POINT: Since the horizontal & vertical motions don’t affect each other we can treat them separately.

How high? need the vertical component of the launch velocity then we can solve it as if we threw the ball straight up at that vertical speed (as we did in Chap 3) figure out how long (time) before the ball comes to a stop, in other words, when is vvertical = 0? then the distance it goes up is: distance up = distance down = y = ½ g t2 = 5 t2

How far? (call this the range of the projectile) need the horizontal component of the launch velocity need to know how long (time) the ball stays in the air (see the how high section to get time) Then since there is no acceleration in the horizontal direction (gravity is in the vertical only) we get that: Range = dacross = x = vhorizontal * t Let’s test our understanding with a battleships question

Launch angle How does launch angle affect the range? Let’s take a trip to the golf range to test things Experiment: Keep the launch speed the same and change the angle to observe the effect. Findings: Maximum range achieved at 45o (no air resistance) also, complimentary angles give same range: 15o and (90o - 15o ) = 75o 30o and (90o - 30o ) = 60o , etc NOTE: air resistance is important, especially for fast moving objects (like baseballs) max range not at 45o when air resistance is taken into account (more about this in the Lab this week)

Example 1 A football player throws a football level to the ground from a height of 1.5 meters. The ball lands 20 meters away from him. How fast was the football going when it left the player’s hand? 1.5 m 20 m

Example 2 A red cross airplane flying level at a speed of 40 m/s must drop relief supplies. If the plane is flying at a height of 500m, how far before the landing site must the plane drop the package? 40 m/s 500 m ?

Example 3 A cannon is fired over level ground at an angle of 30 degrees to the horizontal. The initial velocity of the cannonball is 200 m/s. That means the vertical component of the initial velocity is 100 m/s and the horizontal component is 173 m/s. How long is the cannonball in the air for? How far does the cannonball travel horizontally? Repeat the problem but with a launch angle of 60 degrees. This means the vertical component of the initial velocity is now 173 m/s and the horizontal component is 100 m/s.

Example 4 Cannonball fired: muzzle speed = 141 m/s launch angle = 45o It hits a balloon at top of its trajectory. What is the velocity of the cannonball when it hits the balloon? (Neglect air resistance)

Chapter 6: Momentum

Chapter 6 : Momentum Momentum is inertia (m) in motion (v) momentum = mass * velocity p = m * v UNITS: kg m /s (no special name)

Values of momentum We can get large momentums when: m = 7 kg v = 2 m/s p = 14 kg m /s v = 2 m/s We can get large momentums when: mass is large (supertanker, p = Mv) velocity is large (major league fastball, p = mV) both these are large (Boeing 747, p = MV) m = 0.070 kg p = 14 kg m /s v = 200 m/s

Force & Momentum How do we change momentum? change mass, change velocity or change both momentum = m * v usually keep this same change v So we have acceleration have a net force acting When there is an external force on system, then momentum changes

How force changes momentum F/m = a (Newton’s 2nd Law) now multiply both sides by t and m t * m * F = a * t * m = change in v * t * m m t this leaves us with: Ft = change in (mv) Impulse = change in momentum Racquetball hitting the wall

Changing momentum Increasing momentum: We can consider various changes in momentum and the impulse that produces this change: Increasing momentum decreasing momentum over a long time decreasing momentum over a short time Increasing momentum: When will final velocity be greater: short push or long push? F

Decreasing momentum over a long time: Truck moves with velocity, v. When your brakes fail and you want to stop (v=0) do you: slam into a haystack? slam into a concrete wall? Hint: the change in momentum is same in both cases want to try to minimize the force you feel. Decreasing momentum over a short time: now goal is to maximize force Ex: break a stack of bricks with your hands. Let’s look at a web demo of a car slowing down

Some more examples 1. Which has more momentum: a truck at rest or a dragonfly flying over a pond? 2. A car with a mass of 1000 kg moves at 20m/s. What braking force is needed to bring it to a stop in 10s? ?

Conservation of momentum When a physical quantity remains unchanged during a process, we say that the quantity is conserved So “Conservation of momentum” means that momentum remains unchanged, or Momentum before = momentum after When is momentum conserved? Momentum of a system is conserved when no external forces act on that system Web demo: momentum cart

Example m M Rifle fires a bullet or a cannon fires a cannonball Web demo: cannonball fired Draw situation before the action (firing of rifle): Draw the situation after the firing of the rifle Identify a system on which there are no external forces acting For this system the momentum is conserved: (momentum of system)before = (momentum of system)after m M

More Examples Let’s try an example where a girl jumps off a heavy, stationary cart. As we can guess by now, the cart will move in the opposite direction to the girl and we can figure out how fast if we know a few things. So let’s look at the web demo of the girl jumping off a cart and do some calculations. We can see this same principle at work when a rocket ejects a pellet for propulsion

Now you try one: Two ice skaters are standing still in the middle of the ice when they push off each other. The one skater has a mass of 100 kg, while the other has a mass of 50 kg. If the 100 kg skater has a speed of 2 m/s, what is the speed of the lighter skater? Here’s some questions you should ask yourself as you work through this: Are there any external forces acting? Do you expect the smaller skater to be moving faster or slower than the large skater? Which directions do the skaters move in?

Collisions Elastic collisions (web demo) Inelastic collisions Conservation of momentum is also useful for solving problems involving collisions Elastic collisions (web demo) Colliding objects rebound no deformation of the objects involved Inelastic collisions objects become entangled deformation occurs Perfectly inelastic collisions (web demos) objects stick together after the collision

Using conservation of momentum We can predict the outcome of collisions using conservation of momentum Let’s look at some collisions of carts on an airtrack The objects only exert forces on each other. There are no external forces (like friction) so momentum is conserved: total momentum before the collision = total momentum after the collision

More Inelastic collisions Now lets look at some inelastic collisions and see how mass and speed influence the resulting motions: Big fish/little fish Rear end accident Diesel engine and flatcar Looking at these we are ready to set straight a common movie mistake about momentum:

Bouncing Case A: ball bounces Case B: ball doesn’t bounce A ball of mass 1 kg and travelling at v = 1 m/s hits a wall: Case A: ball bounces before: after: Case B: ball doesn’t bounce In which case is the change in momentum larger? In which case does the wall supply a greater impulse? Let’s first consider a question about bouncing Now let’s look at a bullet hitting a wooden block with and without bouncing.

2-D collisions & explosions In 2D collisions we must take the vector nature of momentum into account: Let’s look at some examples of 2D collisions: 2 objects in an elastic collision 2 cars in an inelastic collision And finally, let’s play some pool/billiards Combined momentum

Chapter 7: Energy

Chap 7: Work and Energy Energy comes in many different forms: energy associated with motion (kinetic) energy associated with position (potential) Chemical energy Heat energy We will focus on mechanical energy: kinetic and potential energies We will consider a number of topics in the Chapter: work done on objects power (rate at which work is done) different types of energy how work and energy are related conservation of energy energy and momentum

Work Last chapter we considered: How long a force is applied Impulse = Force * time Now: How long measured in distance rather than time Work = force * distance W = F * d UNITS: [Joule, J] = [N] . [m] 1 Joule of work is done by a force of 1 Newton exerted on an object over a distance of 1 meter. 1 kiloJoule = 1 kJ = 1000J 1 megaJoule = 1 MJ = 1,000,000 J

Examples Push on a stationary object How much work is done on the object if it remains at rest? Push on a car that moves a distance, d Now pull on the car to slow it down F d F Car moving this way

Example (a) What is the work done on the crate by the rope? A rope applies a horizontal force of 200N to a crate over a distance of 2 meters across the floor. A frictional force of 150 N opposes this motion. (a) What is the work done on the crate by the rope? (b) What is the work done by the frictional force? (c) What is the work done by the support force and the gravitational force on the crate? (d) What is the total work done on the crate?

Work done when lifting objects In order for you to lift something at a constant speed you must exert a force equal to the gravitational force on the object (its weight) You do positive work Gravity does negative work What if the lifting is done at an angle rather than straight up? For instance, if you push a block up an incline? Ans: if you lift the block to the same final height, the work done by gravity is the same in both cases and so the work you do is the same in both cases! What about the force you exert? In which case is it less? Let’s look at an animation of this …..

Power How fast is work being done? Power = work done / time UNITS: [J/s] = [Watt, W] named after James Watt, developer of the steam engine If you do 1 J of work in 1 second, you have used 1 Watt of power. Again we have: 1 kW = 1000W 1 MW = 1,000,000 W

Power calculation examples If little Nellie Newton lifts her 40-kg body a distance of 0.25 meters in 2 seconds, then what is the power delivered by little Nellie's biceps? Two physics students, Albert and Isaac, are in the weightlifting room. Albert lifts the 100-pound barbell over his head 10 times in one minute; Isaac lifts the 100-pound barbell over his head 10 times in 10 seconds. Which student does the most work? Which student delivers the most power?

Other common units of energy Heat energy: In chemistry and in PHY 102 we will use: 1 calorie = 4.19 J Food products: use energy units of: Calories 1 Calorie = 1 kilocalorie = 4190 J Electricity bill: units of energy on bill are: kWhr 1 kWhr = kilowatt * hour = 3,600,000 J [energy] = [power] * [time] A 75 W light bulb uses 75 J of energy per second. If you use the bulb for 4 hours how much energy (in kWhr) do you use?

Another common unit of power Cars measure power in horsepower 1 horsepower = 746 Watts or about 0.75 kiloWatt Origin of the term “horsepower”: Ironically, the term horsepower (hp) was invented by James Watt! He made an estimate of how much work one horse could do in one minute: 33,000 foot-pounds of work / minute. So, for example, a horse exerting 1 hp can raise 330 pounds of coal 100 feet in a minute. Example: My Volvo has a power rating of 175 hp. If I bought it in Sweden how would they advertise the power rating?

Mechanical Energy What enables something to do work? ENERGY we will focus on two types of energy: (1) Kinetic energy (K.E.) energy due to the motion of an object (2) Potential energy (P.E.) energy due to the relative position of an object

Kinetic energy “energy of motion” if v = 0 K.E. = 0 if you push on an object then its velocity increases K.E. increases as well The relationship between the object’s energy and its speed is given by: K.E. = 1/2 * mass * (speed)2 = 1/2 m v2

Examples A dragonfly has mass m = 10 g = 0.01 kg and flies at a speed v = 10 m/s. What is it’s K.E.? A truck has mass m = 2000 kg and v = 2.0 m/s. how much K.E. does the truck have? Determine the kinetic energy of a 1000 kg roller coaster car that is moving with a speed of 20.0 m/s. If the roller coaster car in the above problem were moving with twice the speed, then what would be its new kinetic energy? KE = ? KE = ?

Work & Energy: how are they related? If a force does work on an object it changes the energy of the object Consider a force on a block mass, m, that moves the block a distance, d: Let’s start with Newton’s 2nd Law : Fnet = m a This tells us that the object accelerates If its speed increases, then so does its kinetic energy It can be show that: Wnet = D K.E. Work done on an object = change in object’s K.E. F d Work-energy theorem

Why net work: Wnet? Consider a case where friction is present Fnet = F - f Wnet = Fnet d = D K.E. only part of the work done by the force F goes into changing the bock’s K.E. rest of the energy is transformed into heat energy which results from the friction F f

Example A 1000 kg car moving at 10 m/s (36 km/h) skids 5.0m with locked wheels (wheels not turning) before it stops. How far will the car skid before it stops if it is initially moving at 30 m/s (108 km/h)? We find that: (K.E)case 2 = 9 * (K.E)case 1 but this still doesn’t give us the distance! Work done by the stopping force (brakes) = F * d this does have distance information So we need to use the Work - energy theorem Lets see what this all looks like in action

Potential energy (P.E) Can take on different forms: elastic potential energy: stretched/compressed spring stretched rubber band chemical potential energy in fuels gravitational potential energy P.E. due to elevated position of an object = work done on an object against gravity when lifting it = F * d = (mg) * d (lifting at constant v) = mass * gravity * height P.E. = m * g * h So P.E. is proportional to mass and height. mg F

Where do we measure height, h, from? h is really a change in height must specify a level relative to which we measure h So it is actually better to think of a change in P.E. associated with a change in h In Diagram A we have chosen PE = 0 and h =0 at the bottom. Then all PE values given after that are relative to that reference level. What if we chose instead PE = 0 at the level of the first step in Diagram C. How would the numbers change?

Example How much potential energy does a 100 kg mountaineer gain when they climb Mt Everest (8.84 km) if the mountaineer starts at sea-level? What if the person starts at base camp at 6.0 km? ho = 0 h = 8840 m ho=6000m

Energy continued: Summary Work = Force x distance [N.m = J, Joule] work can be positive (if F and d are in same direction) work can be negative (if F and d are in opposite directions) Work is zero if F and d are perpendicular to each other Power: rate at which work is done = (Work done) / (time interval) [ J/s = W, Watt] Power usage in the home Mechanical energy Kinetic energy: KE = 1/2 m v2 Potential energy gravitational PE = mass x gravity x height

Conservation of Energy Energy can’t be created or destroyed! BUT Energy can be transformed from one form to another Consider some examples: Pendulum: E = P.E. E = K.E. E = P.E. + K.E. E always the same at each position but can be P.E., or K.E. or a combination of both

Diver: see Fig 7.10 Cart on an Incline Projectile Roller coaster Ski jump: Spring potential energy Bungee jumper

What if friction is present? In the previous examples we conserved mechanical energy because we ignored friction and air drag. What if friction is present? Still have conservation of energy but now we conserve total energy instead of mechanical energy Example: When a skier comes down an icy slope (no friction) and then hits a flat section of unpacked snow she slows down. She loses KE because energy is transferred to the snow as heat energy due to friction between her skies and the snow. Demo of this ……

Chemical potential energy More complex systems: Chemical potential energy Radiant energy Provides energy (nutrients) for plant growth ground Plant matter is buried Forms coal Coal is removed we burn it to make electricity

Machines Make life easier (and simpler?) for us again we need to take into account that energy cannot be created or destroyed So: energy in = energy out If there is no friction in the system (ideal machine): work in = work out (F * d )in = (F * d )out Machines can do two things: Fout > Fin (multiply the size of the force) change the direction of the force Example: simple lever can increase the size of the force and change it’s direction

Efficiency Most machines are not ideal This means that (F d )in > (F d )out in other words not everything you put into the machine is available to do work at the output end some of the work you do (i.e. energy you put in) does work against friction we generally consider this to be a “loss” because the output from the machine is diminished of course, the energy is not really lost - it is transformed into a form we don’t find as useful!! Work out = efficiency * (work in)

Example Gasoline contains P.E. = 154 MJ/gallon. Suppose you could build a car with a 100% efficient engine. If at a cruising speed air drag and road friction combine to give you f = 500N, what is your fuel consumption? In other words, how far can you drive on 1 gallon of gas?

Kinetic Energy & Momentum Both momentum & K.E. energy are associated with motion of an object, but: They are not the same type of quantity: Momentum is vector Energy is a scalar What is the relevance of this? Momentum vectors can add together to give you total momentum = 0 Energies can never add up to zero! They have different dependences on velocity Momentum is proportional to v Kinetic energy is proportional to v2

Damage Damage to one object is related to the kinetic energy of the other object striking it. More KE means the striking object can do more work on other object & therefore can deform it more. Example: While playing football you tackle another player in a head-on collision. Your momentums are equal & opposite before the tackle so you come to a dead stop. Question: Which hurts more? To be hit by a fast moving light player?, or A slow moving heavy player? Hint: Which player has more K.E.?

Chapter 8: Rotational Motion

Some questions you’ll be able to answer after today Why is it more fun to be on the outside of the merry-go-round? How do trains go round curves? Why is it easier to balance a hammer when the head is up? Why do SUV’s roll more easily than cars? Why do the clothes in the washer all end up on the outer wall of the washer during the spin cycle? How do ice skaters manage to spin so fast?

Some of these questions stated more scientifically: How do rotational and linear speeds relate? What actually causes things to rotate? About what point do objects rotate? When you put an object down how can you predict whether it will stand or fall over? What is the difference between centripetal force and centrifugal force? If we have linear momentum (mass * velocity) can we have a similar quantity for rotational motion?

Rotational Speed linear speed = distance covered per unit time with circular motion can have constant speed but direction is changing linear speed depends on how far from axis point of interest is Rotational speed = number of rotations per unit time A B

Rotational & Linear speeds Rotational speed usually measured in: RPM = rotations (or revolutions) per minute Example: old vinyl LP RPM = 33 1/3 How are rotational and linear speed related? Linear speed is proportional to: 1. rotational speed, and 2. distance from axis

Turning corners Car wheels Train wheels turn independently outside wheel turns faster than inside one Train wheels fixed axis turns wheels at same rate! but train wheels are tapered Narrow part of the wheel has smaller linear speed (less distance in same amount of time) Wide part of the wheel has a larger linear speed

Rotational Inertia Newton’s 1st Law = Law of inertia Recall: mass is the measure of linear inertia the greater the mass (inertia) of an object the greater the object’s resistance to change in motion (linear acceleration) So: large inertia (mass) small acceleration now rotational inertia (I) is related to mass distribution Ex: most of the mass of the bat is here If you hold bat here large I If you hold bat here smaller I

Rotational Inertia by shape Calculating rotational inertias is tricky, but what we can do is notice that I depends on: (1) shape of the object (2) the axis of rotation you choose Ex: Look at the hoop If most of the mass is located far from axis object has a large I Now just as for linear motion: the greater the rotational inertia of an object the greater the object’s resistance to change in rotational motion (rotational acceleration) So: large I small rotational acceleration

Rotation Why do we get a rotation? Consider linear motion: unbalanced force causes a change in linear motion What is the rotational equivalent? unbalanced torque causes a change in rotational motion Torque = lever arm x force Vector: has magnitude and direction (we will describe direction in terms of the rotation it causes: clockwise or counterclockwise) Distance from the axis of rotation to where the force is applied

Example: seesaw consider the torques exerted by the boy and girl on the seesaw net torque = the sum of the individual torques If net torque = 0 then there are no unbalanced torques and so no rotation! Let’s look at a Web demo Now try Practice Page 31 (mobile)

Center of mass spin something: it seems to rotate about a specific point. Let’s go back to projectile motion: throw a ball and it follows a parabolic path Now throw a baseball, what path does it follow? How does the bat move? What if you spin a wrench across a frictionless table? How does it move?

All these objects rotate about the “center” of the object not a geometric center but rather the: “center of mass” = average position of all the mass that makes up the object Object’s motion can then be separated into: linear motion rotational motion to determine the linear motion of object pretend all the mass of the object is located at the center of mass

Center of mass vs Center of gravity For our purposes: Center of mass (CoM) = Center of gravity (CoG) if gravity (g) is constant everywhere in the object then CoM & CoG are located at the same point CoM & CoG are not in the same location if the object is very large (then g varies across the object)

How do we find the CoG? Let’s consider a few different methods: 1. Symmetric objects: find geometric center if object is symmetric & has uniform density, then: geometric center = center of gravity 2. Find the balance point of the object: 3. Suspend the object: CoG located where the 2 lines cross

Examples Where are the CoG’s located for: Ex 1: Donut Ex 2: L-shape Ex 3: Web Demo: Explorelearning

Stability When will an object stand & when will it topple? Most of us can tell this intuitively but what rule would you give someone? Object is supported by its base See where the CoG is located relative to base: if the CoG is located above the base = stable if the CoG is not over the base = unstable CoG

Circular motion Spin a ball on a string what happens if string snaps? what causes the ball to move in a circle? The string provides a centripetal force What is a centripetal force? Any force that causes circular motion tension force from string gravitational force (moon orbits earth in a circular path) what force keeps a car on a circular track?

Centrifugal force? Centrifugal = center fleeing, away from center So what is this centrifugal force that so many people talk about? Centrifugal = center fleeing, away from center this is an “apparent” force When a car turns corner what happens? The frictional force between car & road causes a centripetal force on car (so the car turns) no seatbelt & slippery seats in your car: you keep going in a straight line it appears as if there is an outward (centrifugal) force acting on you the centrifugal force is actually a lack of a centripetal force on you! Web Link: Right Hand Turn

Example: Amusement park ride you feel like you are pushed outward let’s look at forces acting on you: Spins fast Friction, f Force of wall pushing on you Weight, W = mg This is centripetal force that makes you turn From your perspective you feel an outward: centrifugal force

Angular momentum linear momentum = mass * velocity angular momentum = rotational inertia * rotational velocity = I * w An object’s linear momentum changes only if a force acts on it an object will change its angular momentum only if an external torque acts on it

Conservation of momentum conservation of linear momentum if no external force acts on system then linear momentum is conserved conservation of angular momentum if no external torque acts on system then angular momentum is conserved linear case: (mv)before = (mv)after angular case: (I w)before = (I w)after Web Link: Merry-go-Round

Radians and pi (p) Sometimes angles are measured in degrees Ex: 90o, 45o, etc Can also measure angles in radians [rad] How do we define radians? One complete rotation = 360o circumference of a circle: C = 2 p R if set R = 1, then C = 2 p so distance covered in 1 rotation = 2 p say 2 p radians = 360o or 1 radian = 360o / 2 p R