Conservative Forces

Lecturer:

Professor Stephen T. Thornton

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

A) yes

B) no

Reading Quiz

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

A) yes

B) no

Gravitational PE is Gravitational PE is mghmgh, where height height hh is is measured relative to some arbitrary measured relative to some arbitrary reference level where PE = 0reference 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 levelceiling is the zero level, then the book has negative PE on the tablebook has negative PE on the table. It is only differencesdifferences (or changes) in PE that have any physical meaning.

Last Time

Discussed kinetic energy

Work-energy theorem

TodayConservative and nonconservative forces

Gravitational potential energy

Other kinds of potential energy

Conservation of mechanical energy

Conservation of Energy

• A conservative force does zero total work on

any closed path.

• The work done by a conservative force in

going from an arbitrary point A to an

arbitrary point B is independent of the path

from A to B.

A

B

•

•

•

Doing Work Against Gravity

Energy is reclaimed in this case.

Doing Work Against Gravity

Energy is reclaimed in this case.

Work done by gravity = -mgh

mg

d d

Work Done by Gravity on a Closed Path is Zero.

Work Done by Friction on a Closed Path is Not Zero.

Floor (top view)

kf mg

Conservative ForcesGravitySprings

Nonconservative ForcesFrictionTension

Potential Energy

When we do work, say to lift a box off the floor, then we give the box energy. We call that energy potential energy. Potential energy, in a sense, has potential to do work. It is like stored energy. However, it only works for conservative forces.

Do potential energy demo. Burn string and let large mass drop.

Notes on potential energyPotential energy is part of the work-

energy theorem. Potential energy can be changed into kinetic energy.

Think about gravity for a good example to use.

There is no single “equation” to use for potential energy.

Remember that it is only useful for conservative forces.

Copyright © 2009 Pearson Education, Inc.

Potential EnergyIn raising a mass m to a height h, the work done by the external force is

We therefore define the gravitational potential energy at a height y above some reference point:

.

.

( )ext ext

2 1

F d = W mgh

mg y y

= ×

= -

gravU mgy=

Definition of potential energy

We will (sometimes) use a subscript

on Wc to remind us about conservative

forces. This doesn’t work for friction.

SI unit is the joule (still energy).

( )c i if fW U U U U U

Remember gravityThe work done by a conservative force is equal to the negative of the change in potential energy.

Hold a box up. It has potential energy. Drop the box. Gravity does positive work on the box. The change in the gravitational potential energy is negative. The box has less potential energy when it is on the floor.

More potential energy (PE) notes

Gravitational potential energy = mgh

Only change in potential energy U is important.

There is no absolute value of PE.

We choose the zero of PE to be at the most convenient position to solve problem.

Gravitational potential energy

Because we can choose the “zero” of potential energy anywhere we want, it might be convenient to place it at y = 0 (but not always!).

c

i f c

i f

i f

W mgy

U U U W mgy

U mgy U

U U

iU

fU

y

Where might we choose the zero of potential energy to be here?

Do demos

Loop the loop

Bowling ball (wrecking ball video)

Hopper popper

http://www.youtube.com/watch?v=Rx28g0aqfIk

is B.

An object can have potential energy by virtue of its surroundings.

Familiar examples of potential energy:

• A compressed (or wound-up) spring

• A stretched elastic band

• An object at some height above the ground

Potential EnergyGeneral definition of gravitational potential energy:

For any conservative force:

2

G G

1

U W F dD =- =- ×ò

2

12 1 F dU U U W- ×D = - = =-ò

In one dimension,

We can invert this equation to find F(x) if we know U(x):

In three dimensions:

2Example: 52 5 (other variables remain constant

in partial derivatives)

U xy xyzU

xy xzy

( ) ( )U x F x dx C=- +ò

( )( )

dU xF x

dx=-

( , , )U U U

F x y zx y z

¶ ¶ ¶=- - -

¶ ¶ ¶i j k

Gravitational Potential Energy

Boy does +mgy work to climb up to y. (Gravity does negative work, -mgy). He has potential energy mgy. Gravity does work on boy to bring him down. The potential energy is converted into kinetic energy.

W F d mgy

g

g d

W

m

mgd

Conservation of mechanical energyMechanical energy E is defined to be the sum of K + U.

Mechanical energy is conserved. Only happens for conservative forces.

E K U

Conservation of Mechanical Energy

In the image on the left, the total mechanical energy at any point is:

21

2

E K U

mv mgy

= +

= +

Solving a Kinematics Problem Using Conservation of Energy

E = mgh E = 0

High Jump. In the high jump, the kinetic energy of an athlete is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must the athlete leave the ground in order to lift his center of mass 2.10 m and cross the bar with a speed of 0.70 m/s?

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 float down 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

Conceptual QuizConceptual Quiz

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 float down 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 air resistance exerts a force on it opposite to

its direction of motionits direction of motion. This force does negative workforce does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it fallsleaf loses energy as it falls, and its final total energy is less than its initial total energyfinal total energy is less than its initial total energy.

Conceptual QuizConceptual Quiz

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