Chapter 7 Work and Kinetic Energy

Copyright © 2010 Pearson Education, Inc.

Chapter 7

Work and Kinetic Energy


Reading and Review


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?


Force and Work

N

f

T

mg

displacementAny force not perpendicular

to the motion will do work:

N does no work

T does positive work

f does negative work

mg does negative 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?


Free Fall I

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

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?


Consider the work done by gravity to make the stone

fall distance d:

KE = Wnet = F d cos

KE = mg d

Thus, the stone with the greater mass has the greater

KE, which is twice as big for the heavy stone.

Free Fall I

a) quarter as much

b) half as much

c) the same

d) twice as much


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


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


In the previous question, just

before hitting the ground, what is

the final speed of the heavy stone


a) quarter as much

b) half as much

c) the same

d) twice as much


Free Fall II


In the previous question, just

before hitting the ground, what is

the final speed of the heavy stone


a) quarter as much

b) half as much

c) the same

d) twice as much


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.

Free Fall II


Work Done by a Variable Force

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

Therefore, the work done in stretching the spring is


Application: work by a spring

Hooke’s Law: F = - kx k = (3kg)(9.8 m/s2) / (3.9 cm)k = 760 N/m

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

W = 0.61 J

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

KE = mv2/2 = (1kg)(1m/s)2 / 2KE = 0.55 J

Kinetic Energy:


Power

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

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W

ave

WP

t


Power


Power

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?

F x xP F Fv

t t


a) energy

b) power

c) current

d) voltage

e) none of the above

Electric Bill

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 of power ×

time (watt-hour) is a unit of energy !!

Electric Bill

When you pay the electric

company by the kilowatt-hour,

what are you actually paying for?

a) energy

b) power

c) current

d) voltage

e) none of the above


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 NF = -250 Ny

(b) upper pulley doesn’t move:Tch - 2Trope = 0 Tch = 500 N

lower pulley has constant accelerationTch -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?

What about work?

Trope = 250 NTchain = 500 NF = -250 Ny

(a) P = Fv = 500 N * vbox

(b)P = Fv = 250 N * vhand

hand moves twice as fast

hand moves twice as far


Chapter 8

Potential Energy and Conservation of Energy


Units of Chapter 8

• 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



Work done by gravity on a closed path is zero:



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



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)



Gravitational potential energy:


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.

Sign of the Energy II


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

Question 8.2 KE and PE

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



a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

The gravitational PE depends upon the reference level, but

the difference PE does not! The work done by gravity

must be the same in the two solutions, so PE and KE

should be the same.

Question 8.2 KE and PE


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



Springs: (8-4)


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?



a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

KE and PE




a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

The gravitational PE depends upon the reference level, but

the difference PE does not! The work done by gravity

must be the same in the two solutions, so PE and KE

should be the same.

KE and PE


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.a)Which is travelling faster at the bottom?b)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:



The potential energy curve for a spring:



Contour maps are also a form of potential energy curve:


Summary of Chapter 8

• 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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Chapter 7 Work and Kinetic Energy