14AP Physics C - Work and Energy

7/29/2019 14AP Physics C - Work and Energy

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Work, Energy, and Power

AP Physics C


2/19

There are many different TYPES of

Energy.

Energy is expressed

in JOULES (J)

4.19 J = 1 calorie

Energy can beexpressed more

specifically by using

the term WORK(W)

Work =The Scalar Dot Product between Force and Displacement.So that means if you apply a force on an object and it covers a

displacement you have supplied ENERGY or done WORK on that

object.


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Scalar Dot Product?

A product is obviously a result of

multiplying 2 numbers. A

scalar is a quantity with NO

DIRECTION. So basically

Work is found by multiplying

the Force times the

displacement and result is

ENERGY, which has no

direction associated with it.

nt vectordisplaceme

cos

r

rFrFW

A dot product is basically a CONSTRAINT

on the formula. In this case it means that

F and x MUST be parallel. To ensure that

they are parallel we add the cosine on the

end.

W = Fx

Area = Base x Height


4/19

WorkThe VERTICAL component of the force DOES NOT

cause the block to move the right. The energy imparted to

the box is evident by its motion to the right. Therefore

ONLY the HORIZONTAL COMPONENT of the force

actually creates energy or WORK.

When the FORCE and DISPLACEMENT are in the SAME

DIRECTION you get a POSITIVE WORK VALUE. The

ANGLE between the force and displacement is ZERO

degrees. What happens when you put this in for the

COSINE?

When the FORCE and DISPLACEMENT are in the

OPPOSITE direction, yet still on the same axis, you get a

NEGATIVE WORK VALUE. This negative doesn't mean

the direction!!!! IT simply means that the force and

displacement oppose each other. The ANGLE between the

force and displacement in this case is 180 degrees. What

happens when you put this in for the COSINE?

When the FORCE and DISPLACEMENT are

PERPENDICULAR, you get NO WORK!!! The ANGLE

between the force and displacement in this case is 90

degrees. What happens when you put this in for the

COSINE?


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ExamplecosrFW

rFW

A box of mass m = 2.0 kg is moving over a

frictional floor ( uk = 0.3) has a force whose

magnitude is F = 25 N applied to it at anangle of 30 degrees, as shown to the left.

The box is observed to move 16 meters in

the horizontal direction before falling off thetable.

a) How much work does F do before takingthe plunge?

JW

NmW

W

rFW

rFW

4.346

4.346

30cos1625

cos


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Example contWhat if we had done this in UNIT VECTOR notation?

JWNmW

W

rFrFW

jiF

yyxx

4.3464.346

)05.12()1665.21(

)()(

5.1265.21


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Example cont

JW

FW

rFW

rFW

N

0

90cos16

cos

Fn

How much work does the

FORCE NORMAL do and

Why?

There is NO WORK since

F and r are perpendicular.

Ff

How much work does the frictional force do?

W

W

rFmgW

rFW

rFW

rFW

N

f

f

180cos16)30cos25)8.9(2(3.0

cos)cos(

cos

cos

-34.08 J

Note:This negative does not

specify a direction in this case

since WORK is a SCALAR. It

simply means that the force is

involved in slowing the object

down.


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What if the FORCE IS NOT CONSTANT?

maFNet

dxdt

dvmdxamW

dxmadxFW

)()(

)(

The function here MUST be a FORCE

function with respect to x or r. Lets lookat a POPULAR force function.

Is this function, with respect to x ? NO!

You can still integrate the function, it simply needs to be modified so that it fits

the model accordingly.

v

vo

dvvmW

dvvmdv

dt

dxmW )(


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Work-Energy Theorem

22

)22()|2|(

)(

22

222

o

ov

v

v

v

mvmvW

vvm

vmW

dvvmW

dvvmdvdtdxmW

o

o

KW

mvK

2

21

EnergyKinetic

Kinetic energy is the ENERGY of MOTION.


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Example W=FrcosA 70 kg base-runner begins to slide into second base when moving

at a speed of 4.0 m/s. The coefficient of kinetic friction betweenhis clothes and the earth is 0.70. He slides so that his speed iszero just as he reaches the base (a) How much energy is lostdue to friction acting on the runner? (b) How far does he slide?

)8.9)(70)(70.0(

mgFF nf

= 480.2 N

f

of

f

W

mvW

KWa

22 )4)(70(2

12

10

)

-560 J

x

r

rFW ff

)180(cos)2.480(560

cos

1.17 m


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Another varying force example..

A ball hangs from a rope attached

to a ceiling as shown. A variableforce F is applied to the ball sothat:

F is always horizontal

Fs magnitude varies so that the ball

moves up the arc at a constant

speed.The balls velocity is very low

Assuming the balls mass is m, how much work does F do as it moves from = 0 to = 1?


12/19

Example Cont

tansin)cos

(

sincos

mgmgF

FTmgT

T

mg

Tcos

Tsin

dr

dyor

dx

dy

drmgdrFW

tan

)tan(


13/19

Example Cont

o

o

y

y

y

y

mgymgyW

yymgymgW

dymgdymgW

drdr

dymgdrmgW

o

o

)(||

)(tan

UWmghmgyU

EnergyPotential

The energy of POSITION or

STORED ENERGY is called

Potential Energy!


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Something is missing.

UW

ymgW

yymgW

FrrFW

gravity

gravity

gravity

gravity

180cos)(

cos

21

Consider a mass m that moves from position 1 ( y1)to position 2 m,(y2), moving with a constant velocity.

How much work does gravity do on the body as itexecutes the motion?

Suppose the mass was thrown UPWARD.

How much work does gravity do on thebody as it executes the motion?

UW

ymgW

yymgW

FrrFW

gravity

gravity

gravity

gravity

0cos)(

cos

12

In both cases, the negative

sign is supplied


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The bottom line..The amount of Work gravity does on a body is

PATH INDEPENDANT. Force fields that actthis way are CONSERVATIVE FORCES

FIELDS. If the above is true, the amount of

work done on a body that moves around a

CLOSED PATH in the field will always be

ZERO

FRICTION is a non conservative force. By NON-CONSERVATIVE

we mean it DEPENDS on the PATH. If a body slides up, and then

back down an incline the total work done by friction is NOT ZERO.

When the direction of motion reverses, so does the force and

friction will do NEGATIVE WORK in BOTH directions.


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Force can be found using the DERIVATIVE

dx

dUF

UFxW

Since work is equal to the NEGATIVEchange in potential energy, the

FORCE of an object is the derivative

of the potential energy with respect

to displacement. Be very careful

handling the negative sign.


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Energy is CONSERVED!

afterbefore

oo

oo

oo

EnergyEnergyUKUK

UUKK

UUKKUKW

)(


18/19

ExampleA 2.0 m pendulum is released from rest when the

support string is at an angle of 25 degrees with thevertical. What is the speed of the bob at the bottomof the string?

L

Lcos

h

h = L Lcos

h = 2-2cos

h = 0.187 m

EB = EA

UO = Kmgho = 1/2mv

2

gho = 1/2v2

1.83 = v2

1.35 m/s = v


19/19

How to we measure energy?One of the things we do everyday is measure how

much energy we use. The rate at which we use itdetermines the amount we pay to our utilitycompany. Since WORK is energy the rate at whichwork is done is referred to as POWER.

The unit is either Joules per second or

commonly called the WATT.

To the left are several various versions of

this formula, including some various

Calculus variations.

14AP Physics C - Work and Energy

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