Work, Energy, and Momentum Tanya Liu

Notes All my slides will be uploaded to Professor Dodero’s site:

Some Pronounciations C 1 = “C sub one” or “C one”

Some Pronounciations a + b = a plus b a – b = a minus b a x b = a times b a b = a to the b a 2 = a to the second/ a squared a 3, a 4 = a to the third, a to the fourth, etc

What is work? Work done by a constant force (note the dot product, work can be negative!) F = force on a point Δx = displacement of point of application of force

Dot Product Review = parallel projection of onto If,

What is work?

What is the sign of work done by F friction if F friction is kinetic friction (assume the box is moving to the right)? What is the sign of work done by F friction if F friction is static friction? What is the sign of work done by F person for both cases? F person F Friction

Concept Question: Work What is the work done by the contact force of the wall on the person as the person moves away from the wall? 1. positive 2. negative 3. zero 4. not enough information

Group Question: Work done by Gravity If Bob stands on top of a cliff at a height of y 0 and throws an apple of mass m a downwards, what is the work done by gravity on the falling apple? y = y 0 y = 0

Group Question Solution

Example: Work done by spring force Work: Non-Constant Forces m m x = 0 F spring

Introduction to Energy Definition of energy: – Potential of a physical system to do work Types of energy storage: Kinetic EnergyPotential Energy

Kinetic Energy How is the work done on an object related to the object’s kinetic energy?

Work-Kinetic Energy Theorem Total work done by a net force on an object is equal to the change in kinetic energy of the object

Concept Question: ΔKE of an object Two objects are pushed a distance x from start to finish on a frictionless surface with equal constant forces F. One object has a mass m, while the other has a mass of 4m. Which object has the larger change in kinetic energy? 1. object of mass m 2. object of mass 4m 3. the two objects have the same ΔKE 4. not enough information given x m 4m F F

Concept Question Solution Answer: 3 is correct. The work done on both of the objects is the same, so they will have the same change in KE

Concept Question: ΔKE of an object If both objects start from rest, which object will have the faster velocity at the finish line? 1. object of mass m 2. object of mass 4m 3. the two objects will have the same velocity 4. not enough information given x m 4m F F

Concept Question Solution Answer: 1 is correct. Since ΔKE is the same for both objects,

A ball is attached to a string and is moving in a circle around a post. In case A, the string passes through a hole in the top of the post and is gradually shortened until the ball hits the post. Until the ball hits the post, is the kinetic energy of the ball constant? In case B, the string wraps around the post until it runs out. Is the KE of the ball constant in this case? Concept Question: Peg Wrap A B

Concept Question Solution

Potential Energy Associated with work done by a conservative force – Conservative force is path independent – Gravitational force, spring restoring force Friction is a non-conservative force, depends on path taken

Gravitational Potential Energy yfyf y0y0 m Note: these are defined relative to y=0

Elastic Potential Energy m m x0x0 xfxf Note: these are defined relative to x=0

Concept Question: EPE of a mass There are two objects of mass m on two different springs with spring constants k 1 and k 2. Object 1 is stretched a distance x 1, and object 2 is stretched a distance x 2, with the result that ΔU E1 > ΔU E2. Which of the following could be true? 1.k 1 > k 2 ; x 1 > x 2 2.k 1 > k 2 ; x 1 < x 2 3.k 1 x 2 4.All of the above m m x1x1 k1k1 m m x2x2 k2k2

Concept Question Solution Answer: 4 is correct. All 3 answer choices could give rise to the possibility that ΔU E1 > ΔU E2, as long as either k 1 > k 2 or x 1 > x 2

Conservation of Energy If only conservative forces are acting on an object, we know that: In this case W total = W c, so

Bob loves taking risks, and he wants to skateboard through this vertical loop of radius R. What is the minimum velocity Bob must have at the top to maintain contact with the loop? What height must Bob start at to achieve this velocity? Group Problem: Conservation of energy, conservative forces only R H? V min ?

What happens when we have non- conservative forces? If there are non-conservative forces acting on an object, then and are still true, but now: Conservation of energy with non-conservative forces

Concept Question: Non-conservative forces Back to the man pushing the block What are the non-conservative forces acting on the block, and what is the sign of the work it does on the block?

Concept Question Solution The non-conservative force in this case is friction, and the sign of the work it does is negative.

Group Problem: Conservation of Energy An object of mass m, starting from rest, slides down an inclined plane of length l. The plane is inclined by an angle of 30° to the ground. a.What is the work done by the friction force when the mass is sliding down the ramp? What is the work done by gravity? b.What is the velocity of the block as it reaches the bottom of the ramp? c.How far does the block travel until it finally comes to a rest? (note that the coefficient of kinetic friction is different once the block leaves the ramp) θ = 30°

Group Problem Solution a. b.

Group Problem Solution c.

Work: Non-constant forces But what if F is changing along x? F x xfxf

Work: Non-constant forces What is the work done on an object by the force F as it moves from x=0 to x=x f? F xfxf x

Work: Non-constant forces It is simply the area under the curve F xfxf x Work done by F

Concept Question The following graph shows the force F applied on an object as it moves from x=0 to x=x f. If the object starts from rest and there are no other forces acting on the object, what is the work done on the object? F x xfxf x=1 1 2

Work: Spring Force Spring force is non-constant with displacement x

Group Problem: Work done by Spring Force a.Calculate the work done by the spring force on a spring stretched from x=0 to x=x 2. You must prove your answer using the graph above, simply using the formula given from before is not enough. b.Remember that. Calculate the change in potential energy of the spring if it is stretched from x 1 to x 2. Once again, you must prove your answer using the graph above. Plugging into the given formula is not enough. F x F=-kx x2x2 x1x1

Introduction to Momentum A moving object has momentum, which we define as Momentum of the object Mass of the object Velocity of the object

Conservation of Momentum If there are no external forces acting on a system, we can say that momentum of the system remains constant:

Conservation of Momentum Remember that conservation of momentum applies to a system, so you must carefully define your system when solving momentum problems v1v1 v2v2 System 1 System 3 System 2

Concept Question If the two objects shown below collide with each other on a frictionless surface, in which choice of system is momentum conserved? (What are the external forces on each system?) v1v1 v2v2 System 1 System 3 System 2

Concept Question Answer: System 3. In this system, the force of one block hitting the other is an internal force, so the change in momentum of the system is zero v1v1 v2v2 System 1 System 3 System 2

a.The man and cart move to the right b.The man and cart move to the left c.The man and cart do not move Concept Question: Recoil A man stands on a cart at rest and throws a ball against the wall of the cart. The ball bounces off the wall of the cart in the opposite direction. What will happen to the man and the cart after the ball is thrown?

Answer: b is correct. If we take the man and the cart and the ball as our system, momentum must be conserved meaning Δp has to be 0. The ball moves to the right, so the cart must move to the left Concept Question: Recoil A man stands on a cart at rest and throws a ball against the wall of the cart. The ball bounces off the wall of the cart in the opposite direction. What will happen to the man and the cart after the ball is thrown?

Momentum: Impulse If there is an external force on a system, then its momentum will change

Momentum: Impulse An impulse is a force applied to a system for a certain period of time, Δt Note that these are vectors!

Concept Question: Impulse A small ping pong ball and a massive bowling ball are both rolling towards you with the same momentum. You exert the same amount of force to stop both of them. Which one takes a longer amount of time to stop, and why? a.The time is the same b.The ping pong ball c.The bowling ball

Concept Question

Strategy for Momentum Questions 1.Choose your system 2.Identify initial and final states 3.Identify any external forces to see if momentum is conserved

Group Problem An small block of mass m 1 slides down a circular path of radius R cut into a larger block of mass m 2. The larger block sits on top of a frictionless surface, and both blocks are initially at rest. What is the velocity v 1 of the smaller block just after it leaves the larger block? You will need conservation of momentum and conservation of energy. R m2m2 m1m1 v 1 =?