chapter 8 potential energy and conservation of energy

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Chapter 8

Potential Energy and Conservation of Energy


Reading and Review


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?


Force and Work

N

f

T

mg

displacementAny force not perpendicularto 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 muchb) half as muchc) the samed) twice as muche) 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 muchb) half as muchc) the samed) twice as muche) 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?

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 compared to the light one?

a) quarter as much b) half as muchc) the samed) twice as muche) four times as much

Free Fall II


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

a) quarter as much b) half as muchc) the samed) twice as muche) four times 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


Topics 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 Bb) only Cc) A, B, and Cd) only A and Ce) 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




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.



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



Springs: (8-4)


8-3 Conservation of Mechanical EnergyDefinition of mechanical energy:

(8-6)

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

Or equivalently:


8-3 Conservation of Mechanical EnergyEnergy conservation can make kinematics problems much easier to solve:




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 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:


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


Summary of Chapter 8• 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


Summary of Chapter 8

• 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

chapter 8 potential energy and conservation of energy

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