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Work and Kinetic Energy

Force and Work

a) one force

b) two forces

c) three forces

d) four forces

e) no forces are doing work

A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

N

T

f

mg

Force and Worka) one force

b) two forces

c) three forces

d) four forces

e) no forces are doing work

A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

Any force not perpendicularto the motion will do work:

N doesno work

T doespositivework

f does negative work

mg does negative work

Free Fall I

Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one?

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

Free Fall I

Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one?

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

Consider the work done by gravity to make the stone fall distance d:

DKE = Wnet = F d cosq

DKE = mg d

Thus, the stone with thegreater masshas thegreater

KE, which istwiceas big for the heavy stone.

Follow-up: How do the initial values of gravitational PE compare?

Free Fall II

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one?

Free Fall II

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one?

All freely falling objects fall at the same rate, which is g. Because the acceleration is the same for both, and the distance is the same, then the final speeds will be the same for both stones.

The force needed to stretch a spring an amount x is F = kx.

Therefore, the work done in stretching the spring is

Loaded spring: W = kx2/2= (760 N/m) (0.04m)2/ 2

KE = mv2/2 = (1kg)(1m/s)2 / 2

KE = 0.55 J

Kinetic Energy:

k = (3kg)(9.8 m/s2) / (3.9 cm)

k = 760 N/m

W = 0.61 J

Application: work by a spring

How fast?: v = d/t = (0.020 m) (0.020 s) = 1 m/s

Power is a measure of the rate at which work is done:

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W

If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written:

Question: what is the total work per unit time done on the object?

Electric Bill

a)energy

b) power

c) current

d) voltage

e) none of the above

When you pay the electric company by the kilowatt-hour, what are you actually paying for?

Electric Bill

a)energy

b) power

c) current

d) voltage

e) none of the above

When you pay the electric company by the kilowatt-hour, what are you actually paying for?

We have defined: Power = energy/ time

So we see that:Energy = power× time

This means that the unit ofpower× time (watt-hour) is a unit ofenergy !!

A block rests on a horizontal frictionless surface. A string is attached to the block, and is pulled with a force of 45.0 N at an angle above the horizontal, as shown in the figure. After the block is pulled through a distance of 1.50 m, its speed is 2.60 m/s, and 50.0 J of work has been done on it. (a) What is the angle (b) What is the mass of the block?

The pulley system shown is used to lift a 52 kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine

(a) the force F required to lift the crate with constant speed, and

(b) the tension in two chains

(a) the force F required to lift the crate with constant speed, and

(b) the tension in two chains

(a) constant velocity, a=0, so net force =0.

2T - (52kg)(9.8m/s2) = 0

T = 250 N

F = -250 Ny

(b) upper pulley doesn’t move:

Tch - 2Trope = 0

Tch = 500 N

lower pulley has constant acceleration

Tch -2Trope =0 Tch = 500 N

Mechanical Advantage!

(a) how much power is applied to the box by the chain?

(b) how much power is applied on the rope by the applied force?

Trope = 250 N

Tchain = 500 N

F = -250 Ny

- (a) P = Fv = 500 N * vbox
- P = Fv = 250 N * vhand
- hand moves twice as fast
- hand moves twice as far

Potential Energy and Conservation of Energy

- Conservative and Nonconservative Forces
- Potential Energy and the Work Done by Conservative Forces
- Conservation of Mechanical Energy
- Work Done by Nonconservative Forces
- Potential Energy Curves and Equipotentials

8-1 Conservative and Nonconservative Forces

Conservative force: the work it does is stored in the form of energy that can be released at a later time

Example of a conservative force: gravity

Example of a nonconservative force: friction

Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force

8-1 Conservative and Nonconservative Forces

Work done by gravity on a closed path is zero:

8-1 Conservative and Nonconservative Forces

Work done by friction on a closed path is not zero:

8-1 Conservative and Nonconservative Forces

The work done by a conservative force is zero on any closed path:

8-2 The Work Done by Conservative Forces

If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf; in the meantime, we say the energy is stored as potential energy.

(8-1)

8-2 The Work Done by Conservative Forces

Gravitational potential energy:

Sign of the Energy II

Is it possible for the gravitational potential energy of an object to be negative?

a) yes

b) no

Sign of the Energy II

Is it possible for the gravitational potential energy of an object to be negative?

a) yes

b) no

Gravitational PE is mgh, where height h is measured relative to some arbitrary reference level where PE = 0. For example, a book on a table has positive PE if the zero reference level is chosen to be the floor. However, if the ceiling is the zero level, then the book has negative PE on the table. Only differences (or changes) in PE have any physical meaning.

Question 8.2KE and PE

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

A) skier’s PE B) skier’s change in PE C) skier’s final KE

Question 8.2KE and PE

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

A) skier’s PE B) skier’s change in PE C) skier’s final KE

The gravitational PE depends upon the reference level, but the differenceDPE does not! The work done by gravity must be the same in the two solutions, so DPE and DKE should be the same.

Follow-up: Does anything change physically by the choice of y = 0?

8-3 Conservation of Mechanical Energy

Definition of mechanical energy:

(8-6)

Using this definition and considering only conservative forces, we find:

Or equivalently:

8-3 Conservation of Mechanical Energy

Energy conservation can make kinematics problems much easier to solve:

Example: A mass m slides down a 2 m long smooth ramp which makes an angle of 30o with the top of a table which is 1 m above the floor. The end of the ramp is at the edge of the table. At what horizontal distance from the edge of the table does the mass hit the floor?

KE and PE

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

A) skier’s PE B) skier’s change in PE C) skier’s final KE

KE and PEYou and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

A) skier’s PE B) skier’s change in PE C) skier’s final KE

The gravitational PE depends upon the reference level, but the differenceDPE does not! The work done by gravity must be the same in the two solutions, so DPE and DKE should be the same.

Follow-up: Does anything change physically by the choice of y = 0?

Example: Two water slides are shaped differently, but start at the same height h and are of equal length. Two rides, Paul and Kathy, start from rest at the same time on different slides.

- Which is travelling faster at the bottom?
- Which makes it to the bottom first?

8-4 Work Done by Nonconservative Forces

In 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:

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:

8-5 Potential Energy Curves and Equipotentials

The potential energy curve for a spring:

8-5 Potential Energy Curves and Equipotentials

Contour maps are also a form of potential energy curve:

- Conservative forces conserve mechanical energy
- Nonconservative forces convert mechanical energy into other forms
- Conservative force does zero work on any closed path
- Work done by a conservative force is independent of path
- Conservative forces: gravity, spring

- Work done by nonconservative force on closed path is not zero, and depends on the path
- Nonconservative forces: friction, air resistance, tension
- Energy in the form of potential energy can be converted to kinetic or other forms
- Work done by a conservative force is the negative of the change in the potential energy
- Gravity: U = mgy
- Spring: U = ½ kx2

- Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces
- Nonconservative forces change a system’s mechanical energy
- Work done by nonconservative forces equals change in a system’s mechanical energy
- Potential energy curve: U vs. position

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