Chapter 6 - Giancoli - Work and Energy

Work and EnergyWork and Energy
Mr. SharickMr. Sharick
Giancoli – Chapter 6Giancoli – Chapter 6

Lesson 1: 6.1 - EnergyLesson 1: 6.1 - Energy
BasicsBasics

Lesson 1: EnergyLesson 1: Energy
BasicsBasics
 Energy = Motion (in Mechanics)Energy = Motion (in Mechanics)
 Motion = SpeedMotion = Speed
 The more speed an object has the moreThe more speed an object has the more
energy it “carries.”energy it “carries.”
 Direction doesn’t matter – Energy is aDirection doesn’t matter – Energy is a
scalar, not a vectorscalar, not a vector
 Units for Energy are JoulesUnits for Energy are Joules
 1 Joule = 1 N∙m = 1 kg ∙ m1 Joule = 1 N∙m = 1 kg ∙ m22
/s/s22
(more on this later)(more on this later)

Lesson 1: EnergyLesson 1: Energy
BasicsBasics
 Energy can be transferred to and fromEnergy can be transferred to and from
objects (through acceleration)objects (through acceleration)
 Energy transferred to an object causes theEnergy transferred to an object causes the
object to speed upobject to speed up
 Energy transferred away from an objectEnergy transferred away from an object
causes the object to slow downcauses the object to slow down
 How is this energy transferred?How is this energy transferred?
 Through forces! Remember, forces causeThrough forces! Remember, forces cause
masses to accelerate (F=ma)masses to accelerate (F=ma)

Lesson 1: WORKLesson 1: WORK
 We define this energyWe define this energy
transfer astransfer as workwork..
 WorkWork is a scalaris a scalar
quantity as is energy.quantity as is energy.
 Note the definition ofNote the definition of
work in physics iswork in physics is
different than indifferent than in
common languagecommon language

Lesson 1: WORKLesson 1: WORK
 Work = Energy (in physics)Work = Energy (in physics)
 It is energy that a force produces when theIt is energy that a force produces when the
force pushes or pulls on the mass.force pushes or pulls on the mass.
 If the force is going in the same directionIf the force is going in the same direction
as the motion it is positive work. (givingas the motion it is positive work. (giving
energy to the object)energy to the object)
 If the force is going in the oppositeIf the force is going in the opposite
direction as the motion it is negative work.direction as the motion it is negative work.
 (taking energy from the object)(taking energy from the object)

Mathematical Definition ofMathematical Definition of
Work (Energy) – Dot ProductWork (Energy) – Dot Product
W = the energy that a force transfers to orW = the energy that a force transfers to or
from an object as the object MOVES over afrom an object as the object MOVES over a
distance ∆xdistance ∆x
Work is only done by forces that are in theWork is only done by forces that are in the
direction of or opposing the direction ofdirection of or opposing the direction of
motion.motion.
,( )cos F x
W F x
W F x θ ∆
≡ •∆
= ∆
v v

1-D Example Problems:1-D Example Problems:
1.) A racecar is sitting on the track when a force of 15001.) A racecar is sitting on the track when a force of 1500
N to the East is exerted on it which moves it 10 m to theN to the East is exerted on it which moves it 10 m to the
East. What is the angle between the force F and theEast. What is the angle between the force F and the
displacement Δx? What is the work done by the force?displacement Δx? What is the work done by the force?

1-D Example Problems:1-D Example Problems:
2.) A racecar is moving down the track to the East when a2.) A racecar is moving down the track to the East when a
force of 500 N to the West is exerted on it. The carforce of 500 N to the West is exerted on it. The car
continues to move 20 m while the force is exerted. Whatcontinues to move 20 m while the force is exerted. What
is the angle between the force F and the displacementis the angle between the force F and the displacement
Δx? What is the work done by the force?Δx? What is the work done by the force?

2-D2-D Dot ProductsDot Products
How to take the dot product of any 2 vectors.How to take the dot product of any 2 vectors.
1.1.Draw vectors from a common originDraw vectors from a common origin
2.2.Take angle between vectors. Note: MUST beTake angle between vectors. Note: MUST be
between 0between 0oo
and 180and 180oo
3.3.Multiply to get answer (no direction)Multiply to get answer (no direction)
F = 10 N
Δx = 25 m
θ = 35°
Ex. Prob. 1: What is the work done by the above force?

Total Work (Net Work)Total Work (Net Work)
 The total work done on a particle that movesThe total work done on a particle that moves
from position xfrom position x11 to xto x22 wherewhere
is:is: ( ) ( ) ( )1 2 3
,
...
( )( ) ( )(cos )( )
net
net net x net
W F x F x F x
W F x F xθ
= •∆ + •∆ + •∆ +
= ∆ = ∆
v v vv v v
12 xxx
vvv
−=∆

2-D Example Problem2-D Example Problem
 Calculate the total work done on a mass m asCalculate the total work done on a mass m as
it moves from position xit moves from position x11 = 0 m to x= 0 m to x22 = 40 m= 40 m
X2 = 40X1 = 0
20o
F1 = 5 N
F2 = 6 N
F3 = 2 N
F4 = 2 N
50o

Work ReviewWork Review
QuestionsQuestions
Which of the following statements are true about work?
a. ) Work is a form of energy.
b. ) A Watt is the standard metric unit of work.
c. ) Units of work would be equivalent to a Newton times
a meter.
d.) A kg•m2
/s2
would be a unit of work.

QuestionsQuestions
e.) Work is a time-based quantity; it is dependent upon
how fast a force displaces an object.
f.) Superman applies a force on a truck to prevent it from
moving down a hill. This is an example of work being
done.
g. ) An upward force is applied to a bucket as it is carried
20 m across the yard. This is an example of work being
done.

QuestionsQuestions
h. ) A force is applied by a chain to a roller coaster car to
carry it up the hill of the first drop of the Shockwave ride.
This is an example of work being done.
i. ) The force of friction acts upon a softball player as she
makes a headfirst dive into third base. This is an example
of work being done.
j. ) An eraser is tied to a string; a person holds the string
and applies a tension force as the eraser is moved in a
circle at constant speed. This is an example of work being
done.

QuestionsQuestions
k.) A force acts upon an object to push the object along a
surface at constant speed. By itself, this force must NOT
be doing any work upon the object.
l.) A force acts upon an object at a 90-degree angle to the
direction that it is moving. This force is doing negative
work upon the object.
m.) An individual force does NOT do positive work upon
an object if the object is moving at constant speed.
n.) An object is moving to the right. A force acts leftward
upon it. This force is doing negative work.

Lesson 2: Work andLesson 2: Work and
Kinetic Energy TheoremKinetic Energy Theorem
For a constant net force, the accelerationFor a constant net force, the acceleration
is constant and we can use:is constant and we can use:
I.I.
II.II.
Solve II. forSolve II. for ∆∆x:x:
2 2
2
2
x x
f i x
F ma (Newton's nd Law)
v v a x (Const. Acceleration Equat.)
∑ =
= + ∆
Fx
vi
vf

SubstituteSubstitute ∆∆x into:x into:
This yields:This yields:
2 2
2
f i
x
v v
x
a
−
∆ =
net xW F x≡ ∆

SubstituteSubstitute ∆∆x into:x into:
This yields:This yields:
So work has something to do with theSo work has something to do with the
mass of the object and its speed squared.mass of the object and its speed squared.
2 2
2
f i
x
v v
x
a
−
∆ =
net xW F x≡ ∆
2 2
2 2
( ) ( )
2
1 1
W
2 2
f i
net x
x
net f i
v v
W ma
a
mv mv
−
=
= −

Kinetic EnergyKinetic Energy
 The quantity ½ mvThe quantity ½ mv22
is called Kineticis called Kinetic
Energy (KE)Energy (KE)
 Energy due to motion; if an object hasEnergy due to motion; if an object has
speed it has kinetic energy.speed it has kinetic energy.
21
2
KE mv=

Einstein said this?Einstein said this?
 Einstein’s famous equation is:Einstein’s famous equation is:

Work - Kinetic EnergyWork - Kinetic Energy
TheoremTheorem
 Work done by a constant force is equalWork done by a constant force is equal
to the change of kinetic energy of theto the change of kinetic energy of the
object.object.
 Sign for work designates an increase orSign for work designates an increase or
decrease in energy (Work can bedecrease in energy (Work can be
positive or negative, but KE is alwayspositive or negative, but KE is always
positive; change of KE can be negative)positive; change of KE can be negative)
2 21 1
W
2 2
done f iKE mv mv= ∆ = −

Example: Finding changeExample: Finding change
in KE (work)in KE (work)
A 3kg ball is thrown at a wall withA 3kg ball is thrown at a wall with
The ball loses some energy when it strikes the wall. TheThe ball loses some energy when it strikes the wall. The
ball rebounds with a velocity ofball rebounds with a velocity of
a.) Calculate the change in KE of the particle.a.) Calculate the change in KE of the particle.
b.) If the final velocity of the ball isb.) If the final velocity of the ball is
what would be the net work done on the ball?what would be the net work done on the ball?
xsmvi ˆ/5=

ˆ4.2 /fv m s x= −

ˆ5 /fv m s x=


Lesson 3: UnderstandingLesson 3: Understanding
Potential EnergyPotential Energy

History of Science: Enrique Joule’s
Experiment
•What happens as the masses drop?

History of Science: Enrique Joule’s
Experiment
•What happens as the masses drop?
The original Joule experiment
consists of a receptacle filled with
water and a mechanism with
spinning plates. The kinetic energy
of the plates is transformed into heat,
because the force of gravity performs
work on the weight falling. This gave
an experimental confirmation of the
equivalence between heat and work,
now defined to be exactly 1 calorie
for every 4.184 joules and called a
"thermochemical calorie".
Potential Energy Kinetic Energy Thermal Energy (Heat)

Lesson 3:Lesson 3:
UnderstandingUnderstanding
Potential EnergyPotential Energy Think about when you ride a rollerThink about when you ride a roller
coaster:coaster:
 On the way up energy is beingOn the way up energy is being
transferred to you. (negative worktransferred to you. (negative work
done by gravity)done by gravity)
 On the way down, the energy isOn the way down, the energy is
transferred to kinetic energy (positivetransferred to kinetic energy (positive
work done by gravity; you speed upwork done by gravity; you speed up
going down the hill)going down the hill)

Lesson 3: Work doneLesson 3: Work done
by gravity (potentialby gravity (potential
energy)energy)
As you lift a mass to height h, what is the work done byAs you lift a mass to height h, what is the work done by
gravity?gravity?
yf = h ------------------------
yi =0 m ------------------------

energy)energy)
When the mass falls from the same height h, what is theWhen the mass falls from the same height h, what is the
work done by gravity?work done by gravity?
yf = h ------------------------
yi =0 m ------------------------

 Gravity does negative work when the object isGravity does negative work when the object is
being lifted and the energy is transferred to thebeing lifted and the energy is transferred to the
object (giving it gravitational potential energy)object (giving it gravitational potential energy)
 Change in PE is positive when gravity is doingChange in PE is positive when gravity is doing
negative work (ynegative work (yff > y> yii))
energy)energy)
done by
gravity
f i
PE W
PE mg (y y )
∆ = −
∆ = + −

 Potential energy is defined as a functionPotential energy is defined as a function
of position or height (y) as:of position or height (y) as:
 U is the symbol for potential energyU is the symbol for potential energy
function in some textbooks and collegefunction in some textbooks and college
physics courses. We will stick with usingphysics courses. We will stick with using
PE in this class.PE in this class.
Mathematical PotentialMathematical Potential
Energy function forEnergy function for
GravityGravity
U(y) ≡ mgy

 The value for the potential energy at anyThe value for the potential energy at any
given point due to gravity can thereforegiven point due to gravity can therefore
be found by using the equation:be found by using the equation:
Mathematical PotentialMathematical Potential
Energy function forEnergy function for
GravityGravity
gravityPE mgy=
U(y)
y

Mechanical EnergyMechanical Energy
 The sum of the potential and kinetic energy.
ME = KE + PE
 Mechanical energy in a system can be
conserved if there are ONLY conservative
forces doing work.
Initial energy = final energy
MEi = MEf
KEi + PEi = KEf + PEf
 Mechanical energy in a system will be lost if
there is a non-conservative force doing work.
(more on this later).

Conservative ForcesConservative Forces
 Conservative Forces - Forces for which the work done
only depends on the initial and starting position, but not
the path it takes (gravitational; elastic; electric)
 Ex: Gravity is a conservative force because the work
that gravity does depends only on the change of height
of the object, not the path it takes to get to that height.

Non-conservativeNon-conservative
ForcesForces
Non-conservative Forces – (called dissipative forces)
forces for which the work done depends on the path
taken by the object. (friction, air resistance, tension in
cord, motor or rocket propulsion, push or pull by person)
Ex: The path this box takes as it slides across the floor
matters. It takes the least amount of work (energy) to
slide it in a straight line from point 1 to 2 because you
have to work against a non-conservative force (friction)

EX. A 1kg mass is dropped straight down from aEX. A 1kg mass is dropped straight down from a
height of 20 m. For each of the heights below findheight of 20 m. For each of the heights below find
KE, PE and ME. a.) y = 20 m b.) y = 13 m c.) y = 7 mKE, PE and ME. a.) y = 20 m b.) y = 13 m c.) y = 7 m
d.) y = 0 md.) y = 0 m

 When there is friction present, it “robs” an object ofWhen there is friction present, it “robs” an object of
some of its mechanical energy. The energy is lostsome of its mechanical energy. The energy is lost
to the environment through heat or sound andto the environment through heat or sound and
therefore:therefore:
 Since friction always opposes motion, it alwaysSince friction always opposes motion, it always
does negative work.does negative work.
Lesson 4: ConservationLesson 4: Conservation
of Energy with Frictionof Energy with Friction
(KEi + PEi) ≠ (KEf + PEf)
Specifically:
(KEi + PEi) > (KEf + PEf)
and
Wfr = (KEf + PEf) - (KEi + PEi)

Extending the work-
energy principle
The net work done by a force on an object was found to
be the change in kinetic energy of the object.
Wnet = ΔKE (1)
We write the total (net) work as the sum of the work done
by conservative forces (WC) and the work done by non-
conservative forces (WNC).
Wnet = WNC + WC (2)

Extending the work-
energy principle
Combining eq. 1 and 2 gives:
ΔKE = WNC + WC (3)
Recalling that the work done by a conservative force can
be written as the change in the potential energy (as is the
case for gravity)
Wc = -ΔPE (4)

Extending the work-
energy principle
Solving eq. 3 for WNC yields:
WNC = ΔKE + ΔPE
Thus, the work done by the non-conservative forces
acting on an object is equal to the total change in kinetic
and potential energy.

 A bicyclist (combined mass = 90 kg) starts down a hillA bicyclist (combined mass = 90 kg) starts down a hill
of height 50 m with a velocity of 10 m/s. He coasts untilof height 50 m with a velocity of 10 m/s. He coasts until
coming to a stop 55 meters up the next hill. How muchcoming to a stop 55 meters up the next hill. How much
work did friction do on the bicyclist?work did friction do on the bicyclist?
Lesson 4: ExampleLesson 4: Example
Problem 1Problem 1

Revisiting
Conservative Forces
If only conservative forces are acting, then WNC = 0, thus
ΔKE + ΔPE = 0
Or
(KEf – KEi) + (PEf - PEi) = 0
This means that:
KEf + PEf = KEi + PEi
MEf = MEi
And mechanical energy is conserved! This is true when there are
only conservative forces doing work!

Lesson 4: ExampleLesson 4: Example
Problem 2Problem 2
A ball bearing whose mass, m, is 0.0052A ball bearing whose mass, m, is 0.0052
kg is fired vertically downward from akg is fired vertically downward from a
height, h, of 18m with an initial speed ofheight, h, of 18m with an initial speed of
14 m/s. It buries itself in sand to a depth,14 m/s. It buries itself in sand to a depth,
d, of 0.21 m. What average upwardd, of 0.21 m. What average upward
frictional force does the sand exert on thefrictional force does the sand exert on the
ball as it comes to rest?ball as it comes to rest?

Lesson 5: PowerLesson 5: Power
 Power = the rate at which energy is being usedPower = the rate at which energy is being used
 Average Power:Average Power:
(average over a time interval)(average over a time interval)
 Instantaneous Power:Instantaneous Power:
(at a single instance in time)(at a single instance in time)
W
P
t
dW
P
dt
=
∆
=

Lesson 5: PowerLesson 5: Power
equation 2.0equation 2.0
 Substituting W into power equation:Substituting W into power equation:
 (Constant force)(Constant force)
and
W
P W F x
t
= = •∆
∆
r r
( )
cos
F x
P F v t
t
P Fv θ
•∆
= = •
∆
=
r r r r

Power UnitsPower Units
 SI: WattsSI: Watts
 1 Watt = 1 J/s1 Watt = 1 J/s
 Other units:Other units:
 1 horsepower = 746 Watts1 horsepower = 746 Watts
 British horsepower = 550 ft∙lb/sBritish horsepower = 550 ft∙lb/s
 Applications:Applications:
 Electric energy for U.S. houses is measured inElectric energy for U.S. houses is measured in
kilowatt hours (kWh) which is 3.6 x 10kilowatt hours (kWh) which is 3.6 x 1066
JJ
 This is the amount of energy that a 1000W powerThis is the amount of energy that a 1000W power
source generates in one hour.source generates in one hour.

Power Example: 6.16 –Power Example: 6.16 –
Stair Climbing PowerStair Climbing Power
A 70 kg jogger runs up a long flight of stairs in 4.0s. TheA 70 kg jogger runs up a long flight of stairs in 4.0s. The
vertical height of the stairs is 4.5m.vertical height of the stairs is 4.5m.
a.) Estimate the jogger’s power output in watts anda.) Estimate the jogger’s power output in watts and
horsepower.horsepower.
b.) How much energy did this require?b.) How much energy did this require?

Power Example – 6.59Power Example – 6.59
If a car generates 18 hp (1 hp= 746W) when traveling at aIf a car generates 18 hp (1 hp= 746W) when traveling at a
steady 90 km/hr (25 m/s), what must be the averagesteady 90 km/hr (25 m/s), what must be the average
force exerted on the car due to friction and air resistance?force exerted on the car due to friction and air resistance?

Lesson 6: SpringsLesson 6: Springs

Lesson 6: Hooke’s LawLesson 6: Hooke’s Law
 The minus sign means that the force exerted by theThe minus sign means that the force exerted by the
spring is always opposite the stretching or compressingspring is always opposite the stretching or compressing
motion.motion.
 The spring force is always a restoring force. That is, itThe spring force is always a restoring force. That is, it
wants to restore the mass to the equilibrium position.wants to restore the mass to the equilibrium position.
 The proportionality constant k is called theThe proportionality constant k is called the springspring
constantconstant and is different for different springs. (Large kand is different for different springs. (Large k
means it takes more force to compress or stretch)means it takes more force to compress or stretch)

 Since the force is not constant, but is linear (as seen)Since the force is not constant, but is linear (as seen)
we will use the average force to find the work donewe will use the average force to find the work done
between 2 points xbetween 2 points x11 and xand x22..
by a spring (derive w/by a spring (derive w/
algebra)algebra)

algebra)algebra)
 Since the force is not constant, but is linear (as seen)Since the force is not constant, but is linear (as seen)
we will use the average force to find the work donewe will use the average force to find the work done
between 2 points xbetween 2 points x11 and xand x22..
1 2
1
( )
2
avg x xF F F= +

algebra)algebra)
 The work done is therefore:The work done is therefore:
avgW F x= ∆

algebra)algebra)
1 2 2 1
1
( )( )
2
avg x xW F x F F x x= ∆ = + −

algebra)algebra)
1 2 2 1
1 2 2 1
1
( )( )
2
1
( )( )
2
avg x xW F x F F x x
W kx kx x x
= ∆ = + −
= − − −

algebra)algebra)
1 2 2 1
1 2 2 1
2 2
1 2 2 2 1 1
1
( )( )
2
1
( )( )
2
1
( )
2
W kx kx x x
W kx x kx kx x kx
= ∆ = + −
= − − −
= − − + +

algebra)algebra)
1 2 2 1
1 2 2 1
2 2
1 2 2 2 1 1
2 2
2 1
1
( )( )
2
1
( )( )
2
1
( )
2
1
( )
2
W kx kx x x
W kx x kx kx x kx
W k x x
= ∆ = + −
= − − −
= − − + +
= − −

algebra)algebra)
 Recall that work done by a conservativeRecall that work done by a conservative
force is:force is:
W=-W=-ΔΔPEPE
So:So:
PEPEspringspring= ½ kx= ½ kx22

Lesson 6: ComparingLesson 6: Comparing
PE of spring vs. gravityPE of spring vs. gravity

Lesson 6: Spring ForceLesson 6: Spring Force
problemproblem
A 2 kg block is connected to a spring with a springA 2 kg block is connected to a spring with a spring
constant k=400N/m as it oscillates on a frictionlessconstant k=400N/m as it oscillates on a frictionless
horizontal plane. At x=0, the spring is neither stretchedhorizontal plane. At x=0, the spring is neither stretched
nor compressed.nor compressed.
a.) Determine the work done by the net force on the blocka.) Determine the work done by the net force on the block
as it moves from xas it moves from x11 = 0.1m to x= 0.1m to x22 = 0.2m.= 0.2m.
b.) If the speed of the block is 3 m/s when it is at xb.) If the speed of the block is 3 m/s when it is at x11=0.1m,=0.1m,
how fast is it moving as it passes the point xhow fast is it moving as it passes the point x22 = 0.2m?= 0.2m?

Lesson 6: Work and KELesson 6: Work and KE
for springsfor springs
A 2kg block is moving with a constant speed of 10 m/s on theA 2kg block is moving with a constant speed of 10 m/s on the
horizontal, frictionless plane until it hits the end of the spring. Thehorizontal, frictionless plane until it hits the end of the spring. The
spring constant is 200N/m.spring constant is 200N/m.
a.) How far is the spring compressed before the block comes to resta.) How far is the spring compressed before the block comes to rest
and reverses its motion?and reverses its motion?
b.) What is the speed of the block as the spring subsequently comesb.) What is the speed of the block as the spring subsequently comes
back to its original length?back to its original length?

Chapter 6 - Giancoli - Work and Energy

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