chapter 6 work and energy objectives: the student will be able to: 1.define and calculate...

Chapter 6

Work and Energy

Objectives: The student will be able to:

1.Define and calculate gravitational potential energy.

2.State the work energy theorem and apply the theorem to solve problems.

3.Distinguish between a conservative and a non-conservative force and give examples of each type of force.

6-4 Potential Energy

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

Familiar examples of potential energy:

• A wound-up spring

• A stretched elastic band

• An object at some height above the ground

How is all energy divided?How is all energy divided?

PotentialEnergy

KineticEnergy

All Energy

GravitationPotentialEnergy

ElasticPotentialEnergy

ChemicalPotentialEnergy

What is Potential Energy?What is Potential Energy?

o Energy that is Energy that is stored and waiting stored and waiting to be used laterto be used later

What is Gravitational Potential What is Gravitational Potential Energy?Energy?

o Potential energy Potential energy due to an objectdue to an object’’s s positionposition

o P.E. = mass xP.E. = mass x height x height x

gravitygravity

Don’t look down, Rover!

Good boy!

What is Elastic Potential What is Elastic Potential Energy?Energy?

o Potential energy due compression Potential energy due compression or expansion of an elastic object.or expansion of an elastic object.

Notice the ball compressingand expanding

What is Chemical Potential What is Chemical Potential Energy?Energy?

o Potential energy Potential energy stored within the stored within the chemical bonds chemical bonds of an objectof an object

Which object has more potential energy?Which object has more potential energy?

A

B

ANSWERANSWER

A

This brick has more mass than the feather;therefore more potential energy!

Changing an objectsChanging an objects’’ height can change its height can change its potential energy.potential energy.

If I want to drop an apple from the top of one of these If I want to drop an apple from the top of one of these three things, where will be the most potential energy? three things, where will be the most potential energy?

AB

C

ANSWERANSWER

The higher the The higher the object, the more object, the more potential energy!potential energy!

A

Roller CoastersRoller CoastersWhen does the train on When does the train on this roller coaster have this roller coaster have the MOST potential the MOST potential energy?energy?

AT THE VERY TOP!AT THE VERY TOP!The The HIGHERHIGHER the train is lifted the train is lifted by the motor, the by the motor, the MOREMORE potential energy is produced.potential energy is produced.

At the top of the hill the train At the top of the hill the train has a huge amount of potential has a huge amount of potential energy, but it has very little energy, but it has very little kinetic energy.kinetic energy.


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

We therefore define the gravitational potential energy:

(6-5a)

(6-6)

04/21/23

Potential Energy• Potential energy is associated

with the position of the object• Gravitational Potential Energy is

the energy associated with the relative position of an object in space near the Earth’s surface

• The gravitational potential energy

– m is the mass of an object– g is the acceleration of gravity– y is the vertical position of the mass

relative the surface of the Earth– SI unit: joule (J)

mgyPE

04/21/23

Reference Levels• A location where the gravitational

potential energy is zero must be chosen for each problem– The choice is arbitrary since the change in the

potential energy is the important quantity– Choose a convenient location for the zero

reference height• often the Earth’s surface• may be some other point suggested by the problem

– Once the position is chosen, it must remain fixed for the entire problem

04/21/23

Work and Gravitational Potential Energy

• PE = mgy

•

• Units of Potential Energy are the same as those of Work and Kinetic Energygravity i fW KE PE PE PE

cos ( )cos180

( )

g f i

f i i f

W F y mg y y

mg y y PE PE


This potential energy can become kinetic energy if the object is dropped.

Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).

If , where do we measure y from?

It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.


Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.


The force required to compress or stretch a spring is:

where k is called the spring constant, and needs to be measured for each spring.

(6-8)

Elastic Potential Energy:Learning GoalsThe student will describe the elastic potential energy of a

spring or similar object in qualitative and quantitative terms and will investigate the transformation of gravitational potential to elastic potential.

Elastic Potential Energy

Hooke’s Law

The stretch or compression of an elastic device (e.g. a spring) is directly proportional to the applied force:

Hooke’s Law

The stretch or compression of an elastic device (e.g. a spring) is directly proportional to the applied force:

stre tch xF

kor F kx

x is the equ ilibriu m position

xx

0

The spring constant

The constant k is called the spring constant or force constant. It has units of N/m

and is the slope of the line in a force-extension graph.

Example 1

A student stretches a spring 1.5 cm horizontally by applying a force of magnitude 0.18 N. Determine the force constant of the spring.

Example 1


?

:

18.0

015.0

:

k

Unknown

NF

mx

Givens

Example 1


?

:

18.0

015.0

:

k

Unknown

NF

mx

Givens

mN

m

NkSolve

x

FkkxFSelect

12015.0

18.0:

:


The force stretching or compressing a spring is doing work on a spring, increasing its elastic potential energy. Note that this force is not constant but increases linearly from 0 to kx. The average force on the spring is ½kx.


The force stretching or compressing a spring is doing work on a spring, increasing its elastic potential energy. Note that this force is not constant but increases linearly from 0 to kx. The average force on the spring is ½kx.

W F d F x kx x kx

T h is kx is the elastic po ten tia l energy E

x

e

12

12

2

12

2 , .

Example 2

An apple of mass 0.10 kg is suspended from a vertical spring with spring constant 9.6 N/m. How much elastic potential energy is stored in the spring if the apple stretches the spring 20.4 cm?

?

:

204.0

6.9

:

e

mN

E

Unknown

mx

k

Givens

Example 2

An apple of mass 0.10 kg is suspended from a vertical spring with spring constant 9.6 N/m. How much elastic potential energy is stored in the spring if the apple stretches the spring 20.4 cm?

?

:

204.0

6.9

:

e

mN

E

Unknown

mx

k

Givens

JE

mESolve

kxESelect

e

mN

e

e

20.0

204.06.9:

:2

21

221

Example 2 Follow-Up

How much gravitational potential energy did the apple lose?

Example 2 Follow-Up


?

:

204.0

8.9

10.0

:

2

g

sm

E

Unknown

mh

g

kgm

Givens

Example 2 Follow-Up


?

:

204.0

8.9

10.0

:

2

g

sm

E

Unknown

mh

g

kgm

Givens

JE

mkgESolve

hmgESelect

g

sm

g

g

20.0

204.08.910.0:

:

2

The ideal spring

An ideal spring is one that obeys Hooke’s Law – within compression/stretching limits. Beyond those limits the spring may deform.

The ideal spring

An ideal spring is one that obeys Hooke’s Law – within compression/stretching limits. Beyond those limits the spring may deform.

Be gentle with my springs!

04/21/23

Potential Energy in a Spring• Elastic Potential Energy:

– SI unit: Joule (J)– related to the work required to

compress a spring from its equilibrium position to some final, arbitrary, position x

• Work done by the spring

2

2

1kxPEs

sfsis PEPEW

6-4 Potential EnergyThe force increases as the spring is stretched or compressed further. We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written:

(6-9)

6-5 Conservative and Nonconservative Forces

If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

04/21/23

Types of Forces• Conservative forces

– Work and energy associated with the force can be recovered

– Examples: Gravity, Spring Force, EM forces

• Nonconservative forces– The forces are generally

dissipative and work done against it cannot easily be recovered

– Examples: Kinetic friction, air drag forces, normal forces, tension forces, applied forces …

04/21/23

Conservative Forces• A force is conservative if the work it does

on an object moving between two points is independent of the path the objects take between the points– The work depends only upon the initial and final

positions of the object– Any conservative force can have a potential

energy function associated with it– Work done by gravity– Work done by spring force

fifig mgymgyPEPEW

22

2

1

2

1fisfsis kxkxPEPEW

04/21/23

Nonconservative Forces• A force is nonconservative if the work it

does on an object depends on the path taken by the object between its final and starting points.– The work depends upon the movement path– For a non-conservative force, potential energy

can NOT be defined– Work done by a nonconservative force

– It is generally dissipative. The dispersal of energy takes the form of heat or sound

sotherforceknc WdfdFW


Potential energy can only be defined for conservative forces.


Therefore, we distinguish between the work done by conservative forces and the work done by nonconservative forces.

We find that the work done by nonconservative forces is equal to the total change in kinetic and potential energies:

(6-10)

Applying Potential Energy to Problems

Practice Problem 1By how much does the gravitational potential energy of a 64-kg pole vaulter change if her center of mass rises about 4.0 m during the jump?


Practice Problem 2 #30 in text

A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the person’s head? (c) How is the work done by the person related to the answers in parts (a) and (b)?


A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the person’s head? (c) How is the work done by the person related to the answers in parts (a) and (b)?

(a) Relative to the ground, the PE is given by

2

G book ground 2.10 kg 9.80 m s 2.20 m 45.3 JPE mg y y

b) Relative to the top of the person’s head, the PE is given by

2

G book head 2.10 kg 9.80 m s 0.60 m 12 JPE mg y y h

c) The work done by the person in lifting the book from the ground to the final height is the same as the answer to part (a), 45.3 J. In part (a), the PE is calculated relative to the starting location of the application of the force on the book. The work done by the person is not related to the answer to part (b).

Homework

• Problems in Chapter 6

• #26, 27, 28, 29, 31, 32

Closure

• Kahoot

chapter 6 work and energy objectives: the student will be able to: 1.define and calculate...

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