Work, Energy, and Momentum

Tanya Liu

Notes

• All my slides will be uploaded to Professor Dodero’s site: http://my.liceti.it/prof/dodero/

Some Pronounciations

C1 = “C sub one” or “C one”

Some Pronounciations

• a + b = a plus b• a – b = a minus b• a x b = a times b• ab = a to the b• a2 = a to the second/ a squared• a3, a4 = 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!)

W F x F = force on a point

Δx = displacement of point of application of force

Dot Product Review

= parallel projection of onto

If ,

A B A

B

A

B

cosA

cosA B A B

A B

0A B

What is work?

What is work?

• What is the sign of work done by Ffriction if Ffriction is kinetic friction (assume the box is moving to the right)?

• What is the sign of work done by Ffriction if Ffriction is static friction?

• What is the sign of work done by Fperson for both cases?

Fperson FFriction

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. positive2. negative3. zero4. not enough information

Group Question: Work done by Gravity

If Bob stands on top of a cliff at a height of y0 and throws an apple of mass ma downwards, what is the work done by gravity on the falling apple?

y = y0

y = 0

Group Question Solution

0 0

ˆ

(0 )

jg

g g

F mg

W F y mg y

mg y mgy

Example: Work done by spring force

Work: Non-Constant Forces

m m

x = 0

Fspring

0

2 2 2

0

1 1 1(0)

2 2 2

ˆ

( )f f

spring

x x x x

f

x

xx x

f

F kxi

W F dx kx d

kx

x

kW k x

Introduction to Energy

Definition of energy:– Potential of a physical system to do work

Types of energy storage:

Kinetic Energy Potential Energy

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

2 2, ,0

1

2 x xt tal fo m v vW

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

2 2, ,0

1

2to x f xtal m v vW KE

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 m2. object of mass 4m3. 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 m2. object of mass 4m3. the two objects will have the same velocity4. not enough information given

x

m

4m

F

F

Concept Question Solution

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

1 2

1 2

2 21 1(4 )

2 2

2

mv m v

v v

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

• The path of the ball is not perfectly circular for case A, so the tension force is not perpendicular to the displacement of the ball at all times. Δx ≠ 0, so there is work done on the ball, meaning KE changes

• The path of the ball is always perpendicular to tension force here, so there is no work done, and KE is constant

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

cU W

Gravitational Potential Energy

yf

y0

,0,

0

0

( ( ))

gg f

g g

f

f

UU

U W

mg y y

mgy mgy

m

Note: these are defined relative to y=0

Elastic Potential Energy

m mx0 xf

, ,0

2 2 2 20 0

1 1 1 1

2 2 2 2(

E f E

fE E f

U U

kx kx kx kxU W

Note: these are defined relative to x=0

Concept Question: EPE of a massThere are two objects of mass m on two different springs with spring constants k1 and k2. Object 1 is stretched a distance x1, and object 2 is stretched a distance x2, with the result that ΔUE1 > ΔUE2. Which of the following could be true? 1. k1 > k2 ; x1 > x2

2. k1 > k2 ; x1 < x2

3. k1 < k2 ; x1 > x2

4. All of the above

m m

x1

k1

m m

x2

k2

Concept Question Solution

Answer: 4 is correct. All 3 answer choices could give rise to the possibility that ΔUE1 > ΔUE2, as long as either k1 > k2 or x1 > x2

Conservation of Energy

If only conservative forces are acting on an object, we know that:

In this case Wtotal = Wc, so

KE U

total

c

W KE

W U

0KE U

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

RH?

Vmin ?

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:

totalW KE

total c nc

nc

W W W

KE U W

cW U

ncKE U W 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)

1k2k

θ = 30°

Group Problem Solution

a.

b.

1 1

1 1

1 1 1

0

cos

cos

cos

( ) ( sin ) sin

, f k N N

f k

f f k

g f

F F F mg

F mg

W F x mg l

W mg y y mg l mgl

2 2

1 1

1

1

cos

1( 0 ) cos ( sin )

2

2( sin cos )

, nc

nc f k g

bottom k

bottom k

KE U W

W W mg l U W

m v mg l mgl

v gl g l

Group Problem Solution

c. 0

1

2

121

(0 )2

2

nc

bottom k

bottom

k

KE U W

m v mgx

vx

mg

Work: Non-constant forces

But what if F is changing along x?

W F x F

xxf

Work: Non-constant forces

What is the work done on an object by the force F as it moves from x=0 to x=xf?

F

xf x

Work: Non-constant forces

It is simply the area under the curve

F

xf 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=xf. 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

xxf

x=1

-1

12

Work: Spring Force

Spring force is non-constant with displacement x

springF kx

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=x2. 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 x1 to x2. Once again, you must prove your answer using the graph above. Plugging into the given formula is not enough.

F

x

F=-kx

x2

cW U

x1

Introduction to MomentumA moving object has momentum, which we define as

p mvMomentum 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:

0sysp

Conservation of Momentum

Remember that conservation of momentum applies to a system, so you must carefully define your system when solving momentum problems

v1 v2

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?)

v1 v2

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

v1 v2

System 1

System 3

System 2

a. The man and cart move to the rightb. The man and cart move to the leftc. The man and cart do not move

Concept Question: RecoilA 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: RecoilA 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

ext sys

impulse

F t p

Momentum: Impulse

An impulse is a force applied to a system for a certain period of time, Δt

ext sysF t p

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 sameb. The ping pong ballc. The bowling ball

Concept QuestionA man throws a ball of mass m against a wall with velocity . The ball bounces off the wall in the opposite direction with the same speed as before. Assuming the ball is in contact with the wall for a very short time Δt, which of the following is true about the magnitude of the force F that the wall exerts on the ball?

a. F=2mv/Δtb. F=0c. F=mv/Δt

𝑣 𝑣

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 m1 slides down a circular path of radius R cut into a larger block of mass m2. The larger block sits on top of a frictionless surface, and both blocks are initially at rest. What is the velocity v1 of the smaller block just after it leaves the larger block? You will need conservation of momentum and conservation of energy.

R

m2

m1

v1 =?