Reading and Review. A mass attached to a vertical spring causes the spring to stretch and the mass...

Reading and Review

A mass attached to a vertical

spring causes the spring to

stretch and the mass to move

downwards. What can you

say about the spring’s

potential energy (PEs) and the

gravitational potential energy

(PEg) of the mass?

a) both PEs and PEg decrease

b) PEs increases and PEg decreases

c) both PEs and PEg increase

d) PEs decreases and PEg increases

e) PEs increases and PEg is constant

Question 8.5 Springs and Gravity

A mass attached to a vertical

spring causes the spring to

stretch and the mass to move

downwards. What can you

say about the spring’s

potential energy (PEs) and the

gravitational potential energy

(PEg) of the mass?

a) both PEs and PEg decrease

b) PEs increases and PEg decreases

c) both PEs and PEg increase

d) PEs decreases and PEg increases

e) PEs increases and PEg is constant

The spring is stretched, so its elastic PE increases,

because PEs = kx2. The mass moves down to a

lower position, so its gravitational PE decreases,

because PEg = mgh.

Question 8.5 Springs and Gravity

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Question 8.9 Cart on a Hill

A cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If

the cart were given an initial push, so its

initial speed at the top of the hill was 3 m/s,

what would be its speed at the bottom?

a) 4 m/s

b) 5 m/s

c) 6 m/s

d) 7 m/s

e) 25 m/s

Question 8.9 Cart on a Hill

When starting from rest, thecart’s PE is changed into KE:

PE = KE = m(4)2

A cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If

the cart were given an initial push, so its

initial speed at the top of the hill was 3 m/s,

what would be its speed at the bottom?

a) 4 m/s

b) 5 m/s

c) 6 m/s

d) 7 m/s

e) 25 m/s

When starting from 3 m/s, thefinal KE is:

KEf = KEi + KE

= m(3)2 + m(4)2

= m(25)

= m(5)2

Speed is not the same as kinetic energy

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8-4 Work Done by Nonconservative ForcesIn the presence of nonconservative forces, the total mechanical energy is not conserved:

Solving,

(8-9)

8-4 Work Done by Nonconservative Forces

In this example, the nonconservative force is water resistance:

You see a leaf falling to the ground

with constant speed. When you

first notice it, the leaf has initial

total energy PEi + KEi. You watch

the leaf until just before it hits the

ground, at which point it has final

total energy PEf + KEf. How do

these total energies compare?

a) PEi + KEi > PEf + KEf

b) PEi + KEi = PEf + KEf

c) PEi + KEi < PEf + KEf

d) impossible to tell from

the information provided

Question 8.10a Falling Leaves

You see a leaf falling to the ground

with constant speed. When you

first notice it, the leaf has initial

total energy PEi + KEi. You watch

the leaf until just before it hits the

ground, at which point it has final

total energy PEf + KEf. How do

these total energies compare?

a) PEi + KEi > PEf + KEf

b) PEi + KEi = PEf + KEf

c) PEi + KEi < PEf + KEf

d) impossible to tell from

the information provided

As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy.

Question 8.10a Falling Leaves

Follow-up: What happens to leaf’s KE as it falls? What net work is done?

8-5 Potential Energy Curves and Equipotentials

The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:


The potential energy curve for a spring:


Contour maps are also a form of potential energy curve:

Lecture 11

Linear Momentum

Linear Momentum

Momentum is a vector; its direction is the same as the direction of the velocity.

Going Bowling IGoing Bowling I

p

p

a) the bowling ball

b) same time for both

c) the Ping-Pong ball

d) impossible to say

A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

Going Bowling IGoing Bowling I

p

p

a) the bowling ball




A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

Momentum is p = mv

so the ping-pong ball must have a much greater velocity

Kinetic Energy is KE = 1/2 mv2

so (for a single object): KE = p2 / 2m

Momentum and Newton’s Second Law

Newton’s second law, as we wrote it before:

is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:

Change in Momentum

Change in momentum:

(a) mv

(b) 2mv

A net force of 200 N acts on a 100-

kg boulder, and a force of the same

magnitude acts on a 130-g pebble.

How does the rate of change of the

boulder’s momentum compare to

the rate of change of the pebble’s

momentum?

a) greater than

b) less than

c) equal to

Momentum and ForceMomentum and Force

A net force of 200 N acts on a 100-

kg boulder, and a force of the same

magnitude acts on a 130-g pebble.

How does the rate of change of the

boulder’s momentum compare to

the rate of change of the pebble’s

momentum?

a) greater than

b) less than

c) equal to

The rate of change of momentum is, in fact, the

force. Remember that F = p/t. Because the force

exerted on the boulder and the pebble is the same,

then the rate of change of momentum is the same.

Momentum and ForceMomentum and Force

Impulse

Impulse is a vector, in the same direction as the average force.

The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.

Impulse quantifies the overall change in momentum

Impulse

We can rewrite

as

So we see that

The impulse is equal to the change in momentum.

Why we don’t dive into concreteThe same change in momentum may be

produced by a large force acting for a short time, or by a smaller force acting for a longer time.

Going Bowling IIGoing Bowling II

p

p

a) the bowling ball




A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

Going Bowling IIGoing Bowling II

We know:

Here, F and p are the same for both balls!

It will take the same amount of time to stop them. p

p so p = Fav t

a) the bowling ball




A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

av tp

F

Going Bowling IIIGoing Bowling III

p

p

A bowling ball and a Ping-Pong

ball are rolling toward you with

the same momentum. If you

exert the same force to stop each

one, for which is the stopping

distance greater?

a) the bowling ball

b) same distance for both



Going Bowling IIIGoing Bowling III

p

p

Use the work-energy theorem: W = KE. The ball with less mass has the greater speed, and thus the greater KE. In order to remove that KE, work must be done, where W = Fd. Because the force is the same in both cases, the distance needed to stop the less massive ball must be bigger.

A bowling ball and a Ping-Pong

ball are rolling toward you with

the same momentum. If you

exert the same force to stop each

one, for which is the stopping

distance greater?

a) the bowling ball

b) same distance for both



Conservation of Linear Momentum

The net force acting on an object is the rate of change of its momentum:

If the net force is zero, the momentum does not change!

•A vector equation•Works for each coordinate separately

With no net force:

Internal Versus External Forces

Internal forces act between objects within the system.

As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero:

Therefore, the net force acting on a system is the sum of the external forces acting on it.

Momentum of components of a systemInternal forces cannot change the momentum of a system.

However, the momenta of components of the system may change.

An example of internal forces moving components of a system:

With no net external force:

Kinetic Energy of a SystemAnother example of internal forces moving components of a system:

The initial momentum equals the final (total) momentum.

But the final Kinetic Energy is very large

Opposite case:

Two identical cars travelling at identical

speeds in opposite directions collide head on.

BUT:

VERY inelastic collision!

1, 2, 1, 2,0i i f fp p p p

1, 1, 1, 1, 0i i f fK K K K

Nuclear Fission INuclear Fission I

A uranium nucleus (at rest) A uranium nucleus (at rest)

undergoes fission and splits undergoes fission and splits

into two fragments, one into two fragments, one

heavy and the other light. heavy and the other light.

Which fragment has the Which fragment has the

greater momentum?greater momentum?

a) the heavy one a) the heavy one

b) the light oneb) the light one

c) both have the same c) both have the same

momentummomentum

d) impossible to sayd) impossible to say

11 22

Nuclear Fission INuclear Fission I

A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the

greater momentum?

a) the heavy one

b) the light one

c) both have the same

momentum


1 2

The initial momentum of the uranium

was zero, so the final total momentum

of the two fragments must also be

zero. Thus the individual momenta

are equal in magnitude and opposite

in direction.

Nuclear Fission IINuclear Fission II

a) the heavy one

b) the light one

c) both have the same speed


1 2


greater speed?

Nuclear Fission IINuclear Fission II

We have already seen that the

individual momenta are equal and

opposite. In order to keep the

magnitude of momentum mv the

same, the heavy fragment has the

lower speed and the light fragment

has the greater speed.

a) the heavy one

b) the light one

c) both have the same speed


1 2


greater speed?

Systems with Changing Mass: Rocket Propulsion

If a mass of fuel Δm is ejected from a rocket with speed v, the change in momentum of the rocket is:

The force, or thrust, is

A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece.

We know that px=0, py = 0 in initial stateand no external forces act in the horizontal

An 85-kg lumberjack stands at one end of a 380-kg floating log, as shown in the figure. Both the log and the lumberjack are at rest initially. (a) If the lumberjack now trots toward the other end of the log with a speed of 2.7 m/s relative to the log, what is the lumberjack’s speed relative to the shore? Ignore friction between the log and the water. (b) If the mass of the log had been greater, would the lumberjack’s speed relative to the shore be greater than, less than, or the same as in part (a)? Explain.

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