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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
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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#$, it is able to do work on the blades. *nce
more, mechanical energ# is the abilit# to do work.
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The Total Mechanical Energy
+s alread# mentioned, the mechanical energ#of an object can be the result of its motion
"i.e., kinetic energ#$ 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
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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$?9.
1 Jole = 1 !g • m2"s2
otential Energy
• Definition and Mathematics of Work
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• Calculating the Amount of Work Done by Forces
• Potential Energy
• Kinetic Energy
• Mechanical Energy
• Power
+n object can store energ# as the result of its position. :or e'ample, the
heav# ball of a demolition machine is storing energ# when it is held at an
elevated position. (his stored energ# of position is referred to as potential
energ#. @imilarl#, a drawn bow is able to store energ# as the result of its
position. When assuming its usual position "i.e., when not drawn$, 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
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$ee %ns&er
Elastic Potential Energy
(he second form of potential energ# that we will discuss is elastic potential
energ#. Elastic potential energy is the energ# stored in elastic materials
as the result of their stretching or compressing. 2lastic potential energ# can
be stored in rubber bands, bungee chords, trampolines, springs, an arrow
drawn into a bow, etc. (he amount of elastic potential energ# stored in such
a device is related to the amount of stretch of the device the more stretch,
the more stored energ#.
@prings are a special instance of a device
that can store elastic potential energ# due
to either compression or stretching. + force
is re!uired to compress a spring= the more
compression there is, the more force that is re!uired to compress it further.
:or certain springs, the amount of force is directl# proportional to the amount
of stretch or compression "'$= the constant of proportionalit# is known as the
spring constant "k$.
'spring = ! • (
@uch springs are said to follow ooke%s Law. If a spring is not stretched or
compressed, then there is no elastic potential energ# stored in it. (he spring
is said to be at itsequilibrium position. (he e!uilibrium position is the position
that the spring naturall# assumes when there is no force applied to it. In
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terms of potential energ#, the e!uilibrium position could be called the zero
potential energ# position. (here is a special e!uation for springs that relates
the amount of elastic potential energ# to the amount of stretch "or
compression$ 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
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$ee %ns&er
9. If a force of 1F.G - is used to drag the loaded cart "from previous !uestion$
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
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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
• Calculating the Amount of Work Done by Forces
• Potential Energy
• Kinetic Energy
• Mechanical 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.
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Check Your Understanding
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.
$ee%ns&er
$ee%ns&er
+ 17- frictional force slows a moving block to a stop after a
displacement of 6.7 m to the right.
$ee
%ns&er
$ee
%ns&er
+ 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.
$ee
%ns&er
$ee
%ns&er
+n appro'imatel# 9kg object is sliding at constant speed
across a friction free surface for a displacement of 6 m to the
right.
$ee
%ns&er
$ee
%ns&er
+n appro'imatel# 9kg object is pulled upward at constant
speed b# a 97- force for a vertical displacement of 6 m.
$ee
%ns&er
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%ns&er
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E. /efore beginning its initial descent, a roller coaster car is alwa#s pulled up
the rst hill to a high initial height. Work is done on the car "usuall# b# a
chain$ 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
$ee %ns&er
F. /en (ravlun carries a 977- suitcase up three Oights of stairs "a height of
17.7 m$ 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
$ee %ns&er
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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
$ee %ns&er
N. ow much work is done b# an applied force to lift a 16-ewton block E.7
meters verticall# at a constant speedD
$ee %ns&er
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.
$ee %ns&er
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8. Calculate the work done by a 2.0- !orce (directed at a "0# an$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.)
$ee %ns&er
B. + tired s!uirrel "mass of 1 kg$ 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.
$ee %ns&er
Power
• Definition and Mathematics of Work
• Calculating the Amount of Work Done by Forces
• Potential Energy
• Kinetic Energy
• Mechanical Energy
• Power
(he !uantit# work has to do with a force causing a displacement. Work has nothing
to do with the amount of time that this force acts to cause the displacement.
@ometimes, the work is done ver# !uickl# and other times the work is done rather
slowl#. :or e'ample, a rock climber takes an abnormall# long time to elevate her
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bod# up a few meters along the side of a cli. *n the other hand, a trail hiker "who
selects the easier path up the mountain$ 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.
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+ 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.
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(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.
Check Your Understanding
se #our understanding of work and power to answer the following
!uestions. When nished, click the button to view the answers.