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Mechanical Energy• Definition and Mathematics of Work

• Calculating the Amount of Work Done by Forces

• Potential Energy

• Kinetic Energy

• Mechanical Energy

• Power

In a previous part of Lesson 1, it was said that work is done upon an object

whenever a force acts upon it to cause it to be displaced. Work involves a force

acting upon an object to cause a displacement. In all instances in which work is

done, there is an object that supplies the force in order to do the work. If a World

Civilization book is lifted to the top shelf of a student locker, then the student

supplies the force to do the work on the book. If a plow is displaced across a eld,

then some form of farm e!uipment "usuall# a tractor or a horse$ supplies the force

to do the work on the plow. If a pitcher winds up and accelerates a baseball towards

home plate, then the pitcher supplies the force to do the work on the baseball. If a

roller coaster car is displaced from ground level to the top of the rst drop of a roller

coaster ride, then a chain driven b# a motor supplies the force to do the work on the

car. If a barbell is displaced from ground level to a height above a weightlifter%s

head, then the weightlifter is suppl#ing a force to do work on the barbell. In all

instances, an object that possesses some form of energ# supplies the force to do

the work. In the instances described here, the objects doing the work "a student, a

tractor, a pitcher, a motor&chain$ possess chemical potential energy stored in food

or fuel that is transformed into work. In the process of doing work, the object that is

doing the work e'changes energ# with the object upon which the work is done.

When the work is done upon the object, that object gains energ#. (he energ#ac!uired b# the objects upon which work is done is known as mechanical energy.

)echanical energ# is the energ# that is possessed b# an object due to its

motion or due to its position. )echanical energ# can be either kinetic

energ# "energ# of motion$ orpotential energ# "stored energ# of position$.

*bjects have mechanical energ# if the# are in motion and&or if the# are at

some position relative to a zero potential energy position "for e'ample, a

brick held at a vertical position above the ground or zero height position$. +

moving car possesses mechanical energ# due to its motion "kinetic energ#$.

+ moving baseball possesses mechanical energ# due to both its high speed"kinetic energ#$ and its vertical position above the ground

"gravitational potential energ#$. + World Civilization book at rest on the top

shelf of a locker possesses mechanical energ# due to its vertical position

above the ground "gravitational potential energ#$. + barbell lifted high above

a weightlifter%s head possesses mechanical energ# due to its vertical position

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above the ground "gravitational potential energ#$. + drawn bow possesses

mechanical energ# due to its stretched position

"elastic potential energ#$.

Mechanical Energy as the Ability to DoWork

+n object that possesses mechanical energ# is able

to do work. In fact, mechanical energ# is often

dened as the abilit# to do work. +n# object that possesses mechanical

energ# whether it is in the form of potential energ# orkinetic energ# is

able to do work. (hat is, its mechanical energ# enables that object to appl# a

force to another object in order to cause it to be

displaced.

-umerous e'amples can be given of how an object

with mechanical energ# can harness that energ# in order to appl# a force to

cause another object to be displaced. + classic e'ample involves the massive

wrecking ball of a demolition machine. (he wrecking ball is a massive object

that is swung backwards to a high position and allowed to swing forward into

building structure or other object in order to demolish it. pon hitting the

structure, the wrecking ball applies a force to it in order to cause the wall of

the structure to be displaced. (he diagram below depicts the process b#

which the mechanical energ# of a wrecking ball can be used to do work.

+ hammer is a tool that utilizes mechanical energ# to do work. (he

mechanical energ# of a hammer gives the hammer its abilit# to appl# a force

to a nail in order to cause it to be displaced. /ecause the hammer has











mechanical energ# "in the form of kinetic energ#$, it is able to do work on the

nail. )echanical energ# is the abilit# to do work.

+nother e'ample that illustrates how mechanical

energ# is the abilit# of an object to do work can be

seen an# evening at #our local bowling alle#. (he

mechanical energ# of a bowling ball gives the ball the

abilit# to appl# a force to a bowling pin in order to

cause it to be displaced. /ecause the massive ball has

mechanical energ# "in the form of kinetic energ#$, it is able to do work on the

pin. )echanical energ# is the abilit# to do work.

+ dart gun is still another e'ample of how mechanical energ#

of an object can do work on another object. When a dart gun

is loaded and the springs are compressed, it possesses

mechanical energ#. (he mechanical energ# of the compressed springs gives

the springs the abilit# to appl# a force to the dart in order to cause it to be

displaced. /ecause of the springs have mechanical energ# "in the form of

elastic potential energ#$, it is able to do work on the dart. )echanical energ#

is the abilit# to do work.

+ common scene in some parts of the

countr#side is a 0wind farm.0 ighspeed winds

are used to do work on the blades of aturbine at the socalled wind farm. (he

mechanical energ# of the moving air gives the air particles the abilit# to

appl# a force and cause a displacement of the blades. +s the blades spin,

their energ# is subse!uentl# converted into electrical energ# "a non

mechanical form of energ#$ and supplied to homes and industries in order to

run electrical appliances. /ecause the moving wind has mechanical energ#

"in the form of kinetic energ#$, it is able to do work on the blades. *nce

more, mechanical energ# is the abilit# to do work.











The Total Mechanical Energy

+s alread# mentioned, the mechanical energ#of an object can be the result of its motion

"i.e., kinetic energ#$ and&or the result of its

stored energ# of position "i.e., potential

energ#$. (he total amount of mechanical

energ# is merel# the sum of the potential

energ# and the kinetic energ#. (his sum is simpl# referred to as the total

mechanical energ# "abbreviated ()2$.

TME = PE + KE

+s discussed earlier, there are two forms of potential energ# discussed in ourcourse gravitational potential energ# and elastic potential energ#. 3iven

this fact, the above e!uation can be rewritten4

TME = PEgrav + PEspring + KE

(he diagram below depicts the motion of Li 5ing 5har "esteemed Chinese ski

jumper$ as she glides down the hill and makes one of her recordsetting

jumps.

(he total mechanical energ# of Li 5ing 5har is the sum of the potential and

kinetic energies. (he two forms of energ# sum up to 67 777 8oules. -otice

also that the total mechanical energ# of Li 5ing 5har is a constant value

throughout her motion. (here are conditions under which the total

mechanical energ# will be a constant value and conditions under which it will

be a changing value. (his is the subject of Lesson 9 the workenerg#

relationship. :or now, merel# remember that total mechanical energ# is the

energ# possessed b# an object due to either its motion or its stored energ#

of position. (he total amount of mechanical energ# is merel# the sum of













these two forms of energ#. +nd nall#, an object with mechanical energ# is

able to do work on another object.

inetic energy is the energ# of motion. +n object that has motion whether

it is vertical or horizontal motion has kinetic energ#. (here are man# forms

of kinetic energ# vibrational "the energ# due to vibrational motion$,rotational "the energ# due to rotational motion$, and translational "the

energ# due to motion from one location to another$. (o keep matters simple,

we will focus upon translational kinetic energ#. (he amount of translational

kinetic energ# "from here on, the phrase kinetic energ# will refer to

translational kinetic energ#$ that an object has depends upon two variables4

the mass "m$ of the object and the speed "v$ of the object. (he following

e!uation is used to represent the kinetic energ# ";2$ of an object.

KE = 0.5 • m • v2

where m < mass of object

v < speed of object

(his e!uation reveals that the kinetic energ# of an object is directl#

proportional to the s!uare of its speed. (hat means that for a twofold

increase in speed, the kinetic energ# will increase b# a factor of four. :or a

threefold increase in speed, the kinetic energ# will increase b# a factor of

nine. +nd for a fourfold increase in speed, the kinetic energ# will increase b#

a factor of si'teen. (he kinetic energ# is dependent upon the s!uare of the

speed. +s it is often said, an e!uation is not merel# a recipe for algebraicproblem solving, but also a guide to thinking about the relationship between

!uantities.

;inetic energ# is a scalar !uantit#= it does not have a

direction. nlike velocit#,acceleration, force,

and momentum, the kinetic energ# of an object is

completel# described b# magnitude alone. Like work and potential energ#,

the standard metric unit of measurement for kinetic energ# is the 8oule. +s

might be implied b# the above e!uation, 1 8oule is e!uivalent to 1

kg>"m&s$?9.

1 Jole = 1 !g • m2"s2

otential Energy

• Definition and Mathematics of Work

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• Kinetic Energy


• Power

+n object can store energ# as the result of its position. :or e'ample, the

heav# ball of a demolition machine is storing energ# when it is held at an

elevated position. (his stored energ# of position is referred to as potential

energ#. @imilarl#, a drawn bow is able to store energ# as the result of its

position. When assuming its usual position "i.e., when not drawn$, there is no

energ# stored in the bow. Aet when its position is altered from its usual

e!uilibrium position, the bow is able to store energ# b# virtue of its position.

(his stored energ# of position is referred to as potential energ#. Potential

energy is the stored energ# of position possessed b# an object.

Gravitational Potential Energy

(he two e'amples above illustrate the two forms of potential energ# to be

discussed in this course gravitational potential energ# and elastic potential

energ#. 3ravitational potential energ# is the energ# stored in an object as

the result of its vertical position or height. (he energ# is stored as the result

of the gravitational attraction of the 2arth for the object. (he gravitational

potential energ# of the massive ball of a demolition machine is dependent on

two variables the mass of the ball and the height to which it is raised. (here

is a direct relation between gravitational potential energ# and the mass of an

object. )ore massive objects have greater gravitational potential energ#.

(here is also a direct relation between gravitational potential energ# and the

height of an object. (he higher that an object is elevated, the greater the



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gravitational potential energ#. (hese relationships are e'pressed b# the

following e!uation4

PEgrav = mass • g • height

PEgrav = m #• g • h

In the above e!uation, m represents the mass of the object, h represents the

height of the object and g represents the gravitational eld strength "B.

-&kg on 2arth$ sometimes referred to as the acceleration of gravit#.

(o determine the gravitational potential energ# of an object, a zero height

position mustrst be arbitraril# assigned. (#picall#, the ground is considered

to be a position of zero height. /ut this is merel# an arbitraril# assigned

position that most people agree upon. @ince man# of our labs are done ontabletops, it is often customar# to assign the tabletop to be the zero height

position. +gain this is merel# arbitrar#. If the tabletop is the zero position,

then the potential energ# of an object is based upon its height relative to the

tabletop. :or e'ample, a pendulum bob swinging to and from above the

tabletop has a potential energ# that can be measured based on its height

above the tabletop. /# measuring the mass of the bob and the height of the

bob above the tabletop, the potential energ#

of the bob can be determined.

@ince the gravitational potential energ# of anobject is directl# proportional to its height

above the zero position, a doubling of the

height will result in a doubling of the

gravitational potential energ#. + tripling of the

height will result in a tripling of the

gravitational potential energ#.

se this principle to determine the blanks in the following diagram. ;nowing

that the potential energ# at the top of the tall platform is 67 8, what is the

potential energ# at the other positions shown on the stair steps and the

inclineD



$ee %ns&er

Elastic Potential Energy

(he second form of potential energ# that we will discuss is elastic potential

energ#. Elastic potential energy is the energ# stored in elastic materials

as the result of their stretching or compressing. 2lastic potential energ# can

be stored in rubber bands, bungee chords, trampolines, springs, an arrow

drawn into a bow, etc. (he amount of elastic potential energ# stored in such

a device is related to the amount of stretch of the device the more stretch,

the more stored energ#.

@prings are a special instance of a device

that can store elastic potential energ# due

to either compression or stretching. + force

is re!uired to compress a spring= the more

compression there is, the more force that is re!uired to compress it further.

:or certain springs, the amount of force is directl# proportional to the amount

of stretch or compression "'$= the constant of proportionalit# is known as the

spring constant "k$.

'spring = ! • (

@uch springs are said to follow ooke%s Law. If a spring is not stretched or

compressed, then there is no elastic potential energ# stored in it. (he spring

is said to be at itsequilibrium position. (he e!uilibrium position is the position

that the spring naturall# assumes when there is no force applied to it. In



terms of potential energ#, the e!uilibrium position could be called the zero

potential energ# position. (here is a special e!uation for springs that relates

the amount of elastic potential energ# to the amount of stretch "or

compression$ and the spring constant. (he e!uation is

PEspring = 0.5 • ! • (2

&here ! = spring constant

( = amont o) compression

*relative to eili,rim position-

(o summarize, potential energ# is the energ# that is stored in an object due

to its position relative to some zero position. +n object possesses

gravitational potential energ# if it is positioned at a height above "or below$

the zero height. +n object possesses elastic potential energ# if it is at a

position on an elastic medium other than the e!uilibrium position.

Check Your Understanding

Check #our understanding of the concept of potential energ# b# answering

the following !uestions. When nished, click the button to view the answers.

1. + cart is loaded with a brick and pulled at constant speed along an inclined

plane to the height of a seattop. If the mass of the loaded cart is E.7 kg and

the height of the seat top is 7.F6 meters, then what is the potential energ# of the loaded cart at the height of the seattopD



$ee %ns&er

9. If a force of 1F.G - is used to drag the loaded cart "from previous !uestion$

along the incline for a distance of 7.B7 meters, then how much work is done

on the loaded cartD

$ee %ns&er

-ote that the work done to lift the loaded cart up the inclined

plane at constant speed is e!ual to the potential energ# change of

the cart. (his is not coincidentalH (he reason for the relation between the

potential energ# change of the cart and the work done upon it is the subject

of Lesson 9.

In the rst three units of (he 5h#sics Classroom, we utilized -ewton%s laws to

anal#ze the motion of objects. :orce and mass information were used todetermine the acceleration of an object. +cceleration information was

subse!uentl# used to determine information about the velocit# or

displacement of an object after a given period of time. In this manner,

-ewton%s laws serve as a useful model for anal#zing motion and making

predictions about the nal state of an object%s motion. In this unit, an entirel#

dierent model will be used to anal#ze the motion of objects. )otion will be

approached from the perspective of work and energ#. (he eect that work

has upon the energ# of an object "or s#stem of objects$ will be investigated=

the resulting velocit# and&or height of the object can then be predicted from

energ# information. In order to understand this workenerg# approach to the

anal#sis of motion, it is important to rst have a solid understanding of a few

basic terms. (hus, Lesson 1 of this unit will focus on the denitions and

meanings of such terms as work, mechanical energ#, potential energ#,kinetic

energ#, and power.


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When a force acts upon an object to cause a displacement of the object, it is

said that&or! was done upon the object. (here are three ke# ingredients to

work force, displacement, and cause. In order for a force to !ualif# as

having done work on an object, there must be a displacement and the force

must cause the displacement. (here are several good e'amples of work that

can be observed in ever#da# life a horse pulling a plow through the eld, a

father pushing a grocer# cart down the aisle of a grocer# store, a freshman

lifting a backpack full of books upon her shoulder, a weightlifter lifting a

barbell above his head, an *l#mpian launching the shotput, etc. In each

case described here there is a force e'erted upon an object to cause that

object to be displaced.

Jead the following ve statements and determine whether or not the#

represent e'amples of work. (hen click on the @ee +nswer button to view theanswer.

$tatement %ns&er &ith

E(planation

+ teacher applies a force to a wall and becomes e'hausted.

$ee %ns&er

+ book falls o a table and free falls to the ground.

$ee %ns&er

+ waiter carries a tra# full of meals above his head b# onearm straight across the room at constant speed. "CarefulH

(his is a ver# diKcult !uestion that will be discussed in more

detail later.$

$ee %ns&er

+ rocket accelerates through space.

$ee %ns&er

Work Equation

)athematicall#, work can be e'pressed b# the following e!uation.

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= ' • / • cos

where ' is the force, / is the displacement, and the angle "theta$ is dened

as the angle between the force and the displacement vector. 5erhaps the

most diKcult aspect of the above e!uation is the angle 0theta.0 (he angle is

not just any 'ole angle, but rather a ver# specic angle. (he angle measure is

dened as the angle between the force and the displacement. (o gather an

idea of it%s meaning, consider the

following three scenarios.

• @cenario +4 + force acts

rightward upon an object as itis displaced rightward. In suchan instance, the force vectorand the displacement vectorare in the same direction. (hus, the angle between : andd is 7 degrees.

• @cenario /4 + force acts

leftward upon an object that isdisplaced rightward. In such an instance, the force vector and thedisplacement vector are in the opposite direction. (hus, the anglebetween : and d is 17 degrees.

• @cenario C4 + force acts upward on an object as it is displaced

rightward. In such an instance, the force vector and the displacementvector are at right angles to each other. (hus, the angle between : and dis B7 degrees.

To Do Work !orces Must Cause Dis"lace#ents

Let%s consider @cenario C above in more detail. @cenario C involves a

situation similar to the waiter who carried a tra# full of meals above his head

b# one arm straight across the room at constant speed. It was mentionedearlier that the waiter does not do workupon the tra# as he carries it across

the room. (he force supplied b# the waiter on the tra# is an upward force and

the displacement of the tra# is a horizontal displacement. +s such, the angle

between the force and the displacement is B7 degrees. If the work done b#

the waiter on the tra# were to be calculated, then the results would be 7.

Jegardless of the magnitude of the force and displacement, :>d>cosine B7

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degrees is 7 "since the cosine of B7 degrees is 7$. + vertical force can never

cause a horizontal displacement= thus, a vertical force does not do work on a

horizontall# displaced objectHH

It can be accuratel# noted that the waiter%s

hand did push forward on the tra# for a brief period of time to accelerate it from rest to a

nal walking speed. /ut once up to speed, the

tra# will sta# in its straightline motion at a

constant speed without a forward force. +nd if

the onl# force e'erted upon the tra# during the constant speed stage of its

motion is upward, then no work is done upon the tra#. +gain, a vertical force

does not do work on a horizontall# displaced object.

(he e!uation for work lists three variables each variable is associated with

one of the three ke# words mentioned in thedenition of work "force,displacement, and cause$. (he angle theta in the e!uation is associated with

the amount of force that causes a displacement. +s mentioned in a previous

unit, when a force is e'erted on an object at an angle to the horizontal, onl#

a part of the force contributes to "or causes$ a horizontal displacement. Let%s

consider the force of a chain pulling upwards and rightwards upon :ido in

order to drag :ido to the right. It is onl# the horizontal component of the

tension force in the chain that causes :ido to be displaced to the right. (he

horizontal component is found b# multipl#ing the force : b# the cosine of the

angle between : and d. In this sense, the cosine theta in the work e!uation

relates to the cause factor it selects the portion of the force that actuall#causes a displacement.

The Meaning o$ Theta

When determining the measure of the angle in the work

e!uation, it is important to recognize that the angle has a

precise denition it is the angle between the force and the

displacement vector. /e sure to avoid mindlessl# using any 'ole angle in the e!uation. + common ph#sics lab involves appl#ing a force to

displace a cart up a ramp to the top of a chair or bo'. + force is applied to a

cart to displace it up the incline at constant speed. @everal incline angles are

t#picall# used= #et, the force is alwa#s applied parallel to the incline. (he

displacement of the cart is also parallel to the incline. @ince : and d are in

the same direction, the angle theta in the work e!uation is 7 degrees.

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-evertheless, most students e'perienced the strong temptation to measure

the angle of incline and use it in the e!uation. on%t forget4 the angle in the

e!uation is not just any 'ole angle. It is dened as the angle between the

force and the displacement vector.

The Meaning o$ %egative Work

*n occasion, a force acts upon a moving object to

hinder a displacement. 2'amples might include a car

skidding to a stop on a roadwa# surface or a baseball runner sliding to a stop

on the ineld dirt. In such instances, the force acts in the direction opposite

the objects motion in order to slow it down. (he force doesn%t cause the

displacement but rather hinders it. (hese situations involve what is

commonl# called negative work . (henegative of negative work refers to the

numerical value that results when values of :, d and theta are substituted

into the work e!uation. @ince the force vector is directl# opposite the

displacement vector, theta is 17 degrees. (he cosine"17 degrees$ is 1 and

so a negative value results for the amount of work done upon the object.

-egative work will become important "and more meaningful$ in Lesson 9 as

we begin to discuss the relationship between work and energ#.

Units o$ Work

Whenever a new !uantit# is introduced in ph#sics, the standard metric units

associated with that !uantit# are discussed. In the case of work "and also

energ#$, the standard metric unit is the Jole "abbreviated J$. *ne 8oule is

e!uivalent to one -ewton of force causing a displacement of one meter. In

other words,

The Jole is the nit o) &or!.

1 Jole = 1 e&ton # 1 meter

1 J = 1 # m

In fact, an# unit of force times an# unit of displacement is e!uivalent to a

unit of work. @ome nonstandard units for work are shown below. -otice that





when anal#zed, each set of units is e!uivalent to a force unit times a

displacement unit.

%on&standard Units o$ Work'

foot•pound kg•(m/s2)•m kg•(m2/s2)

In summar#, work is done when a force acts upon an object to cause a

displacement. (hree !uantities must be known in order to calculate the

amount of work. (hose three !uantities are force, displacement and the

angle between the force and the displacement.

(nvestigate)We do work ever# da#. (he work we do consumes Calories ... err, should wesa# 8oules. /ut how much 8oules "or Calories$ would be consumed b# variousactivitiesD se theaily or! widget to investigate the amount of work thatwould be done to run, walk or bike a for a given amount of time at aspecied pace.

Daily Work

Exercise Type (select):

Calculating the Amount of Work Done by Forces• Definition and Mathematics of Work



• Kinetic Energy


• Power

In a previous part of Lesson 1, work was described as taking place when a

force acts upon an object to cause a displacement. When a force acts to

cause an object to be displaced, three !uantities must be known in order tocalculate the work. (hose three !uantities are force, displacement and the

angle between the force and the displacement. (he work is subse!uentl#

calculated as forceMdisplacementMcosine"theta$ where theta is the angle

between the force and the displacement vectors. In this part of Lesson 1, the

concepts and mathematics of work will be applied in order to anal#ze a

variet# of ph#sical situations.
















2'press #our understanding of the concept and mathematics of work b#

answering the following !uestions. When done, click the button to view the

answers.

1. +ppl# the work e!uation to determine the amount of work done b# the

applied force in each of the three situations described below.

$ee %ns&er

9. *n man# occasions, there is more than one force acting upon an object.

+ freebod# diagram is a diagram that depicts the t#pe and the direction of

all the forces acting upon an object. (he following descriptions and their

accompan#ing freebod# diagrams show the forces acting upon an object.

:or each case, indicate which force"s$ are doing work upon the object. (hen

calculate the work done b# these forces.

'ree34o/y 'orces

oing

%mont o)

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Diagram

or!

on the

Object

or! one

by Each

Force

+ 17- force is applied to push a block across a friction freesurface for a displacement of 6.7 m to the right.

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+ 17- frictional force slows a moving block to a stop after a

displacement of 6.7 m to the right.

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+ 17- force is applied to push a block across a frictional

surface at constant speed for a displacement of 6.7 m to the

right.

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+n appro'imatel# 9kg object is sliding at constant speed

across a friction free surface for a displacement of 6 m to the

right.

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+n appro'imatel# 9kg object is pulled upward at constant

speed b# a 97- force for a vertical displacement of 6 m.

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E. /efore beginning its initial descent, a roller coaster car is alwa#s pulled up

the rst hill to a high initial height. Work is done on the car "usuall# b# a

chain$ to achieve this initial height. + coaster designer is considering three

dierent incline angles at which to drag the 9777kg car train to the top of

the N7meter high hill. In each case, the force applied to the car will be

applied parallel to the hill. er critical !uestion is4 which angle would re!uire

the most workD +nal#ze the data, determine the work done in each case, andanswer this critical !uestion.

%ngle 'orce istance or! *J-

a.E6 deg 1.19 ' 17F - 176 m

b.F6 deg 1.EB ' 17F - F.B m

c. 66 deg 1.N1 ' 17F - GE.9 m

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F. /en (ravlun carries a 977- suitcase up three Oights of stairs "a height of

17.7 m$ and then pushes it with a horizontal force of 67.7 - at a constant

speed of 7.6 m&s for a horizontal distance of E6.7 meters. ow much work

does /en do on his suitcase during this entire motionD

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6. + force of 67 - acts on the block at the angle shown in the diagram. (he

block moves a horizontal distance of E.7 m. ow much work is done b# the

applied forceD

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N. ow much work is done b# an applied force to lift a 16-ewton block E.7

meters verticall# at a constant speedD

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G. + student with a mass of 7.7 kg runs up three Oights of stairs in 19.7 sec.

(he student has gone a vertical distance of .7 m. etermine the amount ofwork done b# the student to elevate his bod# to this height. +ssume that her

speed is constant.

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8. Calculate the work done by a 2.0- !orce (directed at a "0# an$le to

the %ertical) to &o%e a '00 $ra& box a horiontal distance o! 00 c&

across a rou$h !loor at a constant speed o! 0.' &*s. (+,T: e cautious

with the units.)

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B. + tired s!uirrel "mass of 1 kg$ does pushups b# appl#ing a force to

elevate its centerofmass b# 6 cm. 2stimate the number of pushups that a

tired s!uirrel must do in order to do a appro'imatel# 6.7 8oules of work.

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Power

• Definition and Mathematics of Work



• Kinetic Energy


• Power

(he !uantit# work has to do with a force causing a displacement. Work has nothing

to do with the amount of time that this force acts to cause the displacement.

@ometimes, the work is done ver# !uickl# and other times the work is done rather

slowl#. :or e'ample, a rock climber takes an abnormall# long time to elevate her















bod# up a few meters along the side of a cli. *n the other hand, a trail hiker "who

selects the easier path up the mountain$ might elevate her bod# a few meters in a

short amount of time. (he two people might do the same amount of work, #et the

hiker does the work in considerabl# less time than the rock climber. (he !uantit#

that has to do with the rate at which a certain amount of work is done is known as

the power. (he hiker has a greater power rating than the rock climber.

5ower is the rate at which work is done. It is the work&time ratio.

)athematicall#, it is computed using the following e!uation.

Po&er = or! " time

or

P = " t

(he standard metric unit of power is the att. +s

is implied b# the e!uation for power, a unit of

power is e!uivalent to a unit of work divided b# a

unit of time. (hus, a Watt is e!uivalent to a

8oule&second. :or historical reasons,

the horsepower is occasionall# used to describe the power delivered b# a

machine. *ne horsepower is e!uivalent to appro'imatel# G67 Watts.

)ost machines are designed and built to do work on objects. +ll machines

are t#picall# described b# a power rating. (he power rating indicates the rate

at which that machine can do work upon other objects. (hus, the power of a

machine is the work&time ratio for that particular machine. + car engine is an

e'ample of a machine that is given a power rating. (he power rating relates

to how rapidl# the car can accelerate the car. @uppose that a F7horsepower

engine could accelerate the car from 7 mi&hr to N7 mi&hr in 1N seconds. If thiswere the case, then a car with four times the horsepower could do the same

amount of work in onefourth the time. (hat is, a 1N7horsepower engine

could accelerate the same car from 7 mi&hr to N7 mi&hr in F seconds. (he

point is that for the same amount of work, power and time are inversel#

proportional. (he power e!uation suggests that a more powerful engine can

do the same amount of work in less time.



+ person is also a machine that has a power rating.

@ome people are more powerfull than others. (hat is,

some people are capable of doing the same amount of

work in less time or more work in the same amount of

time. + common ph#sics lab involves !uickl# climbing a Oight of stairs and

using mass, height and time information to determine a student%s personal

power. espite the diagonal motion along the staircase, it is often assumed

that the horizontal motion is constant and all the force from the steps is used

to elevate the student upward at a constant speed. (hus, the weight of the

student is e!ual to the force that does the work on the student and the

height of the staircase is the upward displacement. @uppose that /en

5umpiniron elevates his 7kg bod# up the 9.7meter stairwell in 1.

seconds. If this were the case, then we could calculate /en%s power rating. It

can be assumed that /en must appl# an 77-ewton downward force upon

the stairs to elevate his bod#. /# so doing, the stairs would push upward on/en%s bod# with just enough force to lift his bod# up the stairs. It can also be

assumed that the angle between the force of the stairs on /en and /en%s

displacement is 7 degrees. With these two appro'imations, /en%s power

rating could be determined as shown below.

/en%s power rating is G1 Watts. e is !uite a horse.

Another !or#ula $or Po*er

(he e'pression for power is work&time. +nd since the e'pression for work is

force>displacement, the e'pression for power can be rewritten as

"force>displacement$&time. @ince the e'pression for velocit# is

displacement&time, the e'pression for power can be rewritten once more as

force>velocit#. (his is shown below.



(his new e!uation for power reveals that a

powerful machine is both strong "big force$ and

fast "big velocit#$. + powerful car engine is strong

and fast. + powerful piece of farm e!uipment is

strong and fast. + powerful weightlifter is strongand fast. + powerful lineman on a football team

is strong and fast. + machine that is strong

enough to appl# a big force to cause a displacement in a small mount of time

"i.e., a big velocit#$ is a powerful machine.


se #our understanding of work and power to answer the following

!uestions. When nished, click the button to view the answers.

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