Download Work- Energy Theorem

TOPIC 5:
Work, Energy &
Power
WORK

Definition of Work:
When a force causes a displacement of an
object
 Components of the force need to be in the
direction of the displacement

Net Work done by a
Constant Net Force
Work = Force (F) x Displacement (x)
W = Fx
W = Fx = (Fcosθ)x
** Only the component of the force in the direction of the displacement,
contributes to work
Units of Work
Work = Force x Displacement
= Newtons x meters
Newton x meter  Joule (J)
* Joule is named after James Prescott Joule (18181889) who made major contributions to the
understanding of energy, heat, and electricity
Work

Work:
Scalar quantity
 Can be positive or negative
 Positive work  Exists when the force &
displacement vectors point in the same
direction
 Negative work  Exists when the force &
displacement vectors point in opposite
directions

Problem
How much work is done on a vacuum
cleaner pulled 3 m by a force of 50 N at
an angle of 30° above the horizontal?
W = (Fcosθ)x W = ?
F = 50N
d = 3m θ = 30°
W = (50N)(cos30°)(3m)
= 130 J
ENERGY
Kinetic Energy:
* Energy associated with an object in motion
* Depends on speed and mass
* Scalar quantity
* SI unit for all forms of energy = Joule (J)
KE = ½ mv2
KE = ½ x mass x (velocity)2
Kinetic Energy
If a bowling ball and a soccer ball are
traveling at the same speed, which do you
think has more kinetic energy?
KE = ½ mv2
* Both are moving with identical speeds
* Bowling ball has more mass than the soccer ball 
Bowling ball has more kinetic energy
Kinetic Energy Problem

A 7 kg bowling ball moves at 3 m/s. How
fast must a 2.45 g tennis ball move in
order to have the same kinetic energy as
the bowling ball?
Velocity of tennis ball = 160 m/s
Work-Kinetic Energy Theorem
Work-kinetic Energy Theorem:
•
Net work done on a particle equals the change
in its kinetic energy (KE)
W = ΔKE
W  KEf  KEo  mv  mv
1
2
2
f
1
2
2
o
PROBLEM

What is the soccer ball’s speed
immediately after being kicked? Its
mass is 0.42 kg.
PROBLEM

What is the soccer ball’s speed
immediately after being kicked? Its
mass is 0.42 kg.
W = F ∙ Δx
W = (240 N) (0.20 m) = 48 J
W = ΔKE = 48 J
KE = ½ mv2 = 48 J
v2 = 2(48 J)/0.42 kg
v = 15 m/s
Work-Kinetic Energy Theorem
On a frozen pond, a person kicks a 10 kg sled,
giving it an initial speed of 2.2 m/s. How far
does the sled move if the coefficient of kinetic
friction between the sled and the ice is 0.10?
m = 10 kg vi = 2.2 m/s vf = 0 m/s μk = 0.10
d=?
Work-Kinetic Energy Theorem
Wnet = Fnetdcosθ
* Net work done of the sled is provided by the force of
kinetic friction
Wnet = Fkdcosθ  Fk = μkN  N = mg
Wnet = μkmgdcosθ
* The force of kinetic friction is in the direction opposite
of d  θ = 180°
* Sled comes to rest  So, final KE = 0
Wnet = Δ KE = ½ mv2f – ½ mv2i
Wnet = -1/2 mv2i
Work-Kinetic Energy Theorem
Use the work-kinetic energy theorem, and
solve for d
Wnet = ΔKE
- ½ mv2i = μkmgdcosθ
d = 2.5 m
POWER
POWER:
* A quantity that measures the rate at which
work is done or energy is transformed
* Power = work / time interval
P = W/Δt
(W = Fx  P = Fx/Δt  v = x/Δt)
* Power = Force x speed
P = Fv
POWER
SI Unit for Power:
Watt (W)  Defined as 1 joule per second (J/s)
Horsepower = Another unit of power
1 hp = 746 watts
POWER PROBLEM
A 193 kg curtain needs to be raised 7.5 m,
in as close to 5 s as possible. The power
ratings for three motors are listed as 1
kW, 3.5 kW, and 5.5 kW. What motor is
best for the job?
POWER PROBLEM
m = 193 kg Δt = 5s
d =7.5m
P=?
P = W/Δt
= Fx/Δt
= mgx/Δt
= (193kg)(9.8m/s2)(7.5m)/5s
= 280 W  2.8 kW
** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the
curtain fast enough, and the 5.5 kW motor will lift the curtain too fast
POTENTIAL ENERGY
Potential Energy:
* Stored energy
* Associated with an object that has the
potential to move because of its position relative
to some other location
Example:
Balancing rock- Arches National Park, Utah
Delicate Arch- Arches National Park, Utah
GRAVITATIONAL POTENTIAL ENERGY- Definition
Gravitational potential energy PEg is the energy an
object of mass m has by virtue of its position
relative to the surface of the earth. That position
is measured by the height h of the object relative
to an arbitrary zero level:
PEg = mgh
SI Unit = Joule (J)
Problem

What is the bucket’s gravitational
potential energy?
Problem

What is the bucket’s gravitational
potential energy?
PE = mgh
PE = (2.00 kg)(9.80 m/s2)(4.00 m)
PE = 78.4 J
Gravitational Potential Energy
Example: A Gymnast on a Trampoline
The gymnast leaves the trampoline at an initial height of 1.20 m
and reaches a maximum height of 4.80 m before falling back
down. What was the initial speed of the gymnast?
Gravitational Potential Energy
W  12 mvf2  12 mvo2
mgho  h f    12 mvo2
Wgravity  mg ho  h f 
vo   2 g ho  h f 


vo   2 9.80 m s 2 1.20 m  4.80 m   8.40 m s
Elastic Potential Energy
* Energy stored in any compressed or stretched
object

Spring, stretched strings of a tennis racket or guitar, rubber
bands, bungee cords, trampolines, an arrow drawn into a
bow, etc.
Springs

When an external force compresses or stretches
a spring  Elastic potential energy is stored in
the spring


The more stretch, the more stored energy
For certain springs, the amount of force is directly
proportional to the amount of stretch or compression
(x);

Constant of proportionality is known as the spring constant (k)
Fspring = k * x
Hooke’s Law

If a spring is not stretched or compressed
 no potential energy is being stored

Spring is in an Equilibrium position

Equilibrium position: Position spring naturally
assumes when there is no force applied to it

Zero potential energy position
Hooke’s Law

Special equation for springs

Relates the amount of elastic potential
energy to the amount of stretch (or
compression) and the spring constant
PE elastic = ½kx2
k = Spring constant (N/m)
Stiffer the spring  Larger the spring constant
x = Amount of compression relative to the equilibrium
position
Potential Energy Problem
A 70 kg stuntman is attached to a bungee cord with an
unstretched length of 15 m. He jumps off the bridge
spanning a river from a height of 50m. When he finally
stops, the cord has a stretched length of 44 m. Treat the
stuntman as a point mass, and disregard the weight of
the bungee cord. Assuming the spring constant of the
bungee cord is 71.8 N/m, what is the total potential
energy relative to the water when the man stops falling?
Potential Energy Problem
* Zero level for gravitational potential energy is chosen
to be the surface of the water
* Total potential energy  sum of the gravitational &
elastic potential energy
PEtotal = PEg + PEelastic
= mgh + ½ kx2
* Substitute the values into the equation
PEtotal = 3.43 x 104 J
Potential Energy

The energy stored in an object due to its
position relative to some zero position
An object possesses gravitational potential
energy if it is positioned at a height above (or
below) the zero height
 An object possesses elastic potential energy if
it is at a position on an elastic medium other
than the equilibrium position

Linking Work to Mechanical
Energy



WORK is a force acting upon an object to
cause a displacement
When work is done upon an object, that
object gains energy
Energy acquired by the objects upon
which work is done is known as
MECHANICAL ENERGY
Mechanical Energy

Objects have mechanical energy if they
are in motion and/or if they are at some
position relative to a zero potential energy
position
Total Mechanical Energy
*Total Mechanical Energy: The sum of
kinetic energy & all forms of potential energy
1. Kinetic Energy (Energy of motion)
KE = ½ mv2
2. Potential Energy (Stored energy of
position)
a. Gravitational
PEg = mgh
b. Elastic
2
Mechanical Energy
CONSERVATION OF MECHANICAL ENERGY:
* In the absence of friction, mechanical energy is
conserved, so the amount of mechanical energy remains
constant
MEi = MEf
Initial mechanical energy = final mechanical energy
(in the absence of friction)
PEi + KEi = PEf + KEf
mghi + ½ mvi2 = mghf + ½ mvf2
Conservation of Energy
Problem
Starting from rest, a child zooms down a
frictionless slide from an initial height of 3
m. What is her speed at the bottom of the
slide? (Assume she has a mass of 25 kg)
Conservation of Energy
Problem
hi = 3m
hf = 0m
•
•
•
vi = 0 m/s
Slide is frictionless  Mechanical energy is conserved
Kinetic energy & potential energy = only forms of energy
present
•
•
m = 25kg
vf = ?
KE = ½ mv2
PEg = mgh
Final gravitational potential energy = zero (Bottom of
the slide)  PEgf = 0
Initial gravitational potential energy  Top of the slide
 PEgi = mghi  (25kg)(9.8m/s2)(3m) = 736 J
Conservation of Energy
Problem
hi = 3m
hf = 0m
•
KEi = 0
Final Kinetic Energy
•
•
vi = 0 m/s
Initial Kinetic Energy = 0, because child starts at rest
•
•
m = 25kg
vf = ?
KEf = ½ mv2  ½ (25kg)v2f
MEi = MEf
PEi + KEi = PEf + Kef
736 J + 0 J = 0 J + (1/2)(25kg)(v2f)
vf = 7.67 m/s
Mechanical Energy  Ability
to do Work

An object that possesses mechanical
energy is able to do work
Its mechanical energy enables that object to apply
a force to another object in order to cause it to be
displaced
 Classic Example  Massive wrecking ball of a
demolition machine

Mechanical Energy is the
ability to do work…


An object that possesses mechanical
energy (whether it be kinetic energy or
potential energy) has the ability to do
work
That is… its mechanical energy enables
that object to apply a force to another
object in order to cause it to be displaced
Mechanical Energy
Work is a force acting on an object to
cause a displacement
 In the process of doing work  the object
which is doing the work exchanges energy
with the object upon which the work is
done
 When work is done up the object  that
object gains energy

Mechanical Energy

A weightlifter applies a force to cause a
barbell to be displaced

Barbell now possesses mechanical energy- all
in the form of potential energy
** The energy acquired by the objects
upon which work is done is known as
mechanical energy
Mechanical Energy is the
ability to do work…
Examples on website:
Massive wrecking ball of a demolition machine
The wrecking ball is a massive object which is
swung backwards to a high position and allowed to
swing forward into a building structure or other object in
order to demolish it
Upon hitting the structure, the wrecking ball applies
a force to it in order to cause the wall of the structure to
be displaced
Mechanical energy = ability to do work
Work- Energy Theorem
Categorize forces based upon whether or not
their presence is capable of changing an object’s
total mechanical energy
* Certain types of forces, which when present and
when involved in doing work on objects, will change the
total mechanical energy of the object
* Other types of forces can never change the total
mechanical energy of an object, but rather only
transform the energy of an object from PE to KE or vice
versa
** Two categories of forces  Internal & External
Work- Energy Theorem
External Forces:
Applied force, normal force, tension force, friction
force and air resistance force
Internal Forces:
Gravity forces, spring forces, electrical forces and
magnetic forces
Work- Energy Theorem
THE BIG CONCEPT!!
* When the only type of force doing net work upon an
object is an internal force (gravitational and spring
forces)
 Total mechanical energy (KE + PE) of that object
remains constant
 Object’s energy simply changes form 
Conservation of Energy
** Ex) As an object is “forced” from a high elevation to a lower
elevation by gravity  Some of the PE is transformed into KE (Yet,
the sum of KE + PE = remains constant)
Work- Energy Theorem
THE BIG CONCEPT!!
* If only internal forces are doing work  energy
changes forms (KE to PE or vice versa)  total
mechanical energy is therefore conserved
* Internal forces – referred to as conservative forces
Quick Quiz
Work-Energy Relationship
Analysis of situations in which work is conserved  only
internal forces are involved
TMEi + WEXT = TMEf
(Initial amount of total mechanical energy (TMEi) plus the work done by external
forces (WEXT)  equals the final amount of total mechanical energy (TMEf))
KEi + PEi + Wext = KEf + PEf
KEi + PEi = KEf + Pef
Website
Work- Energy Theorem
THE BIG CONCEPT!!
* Forces are categorized as being either internal or
external based upon the ability of that type of force to
change an object’s total mechanical energy when it does
work upon an object
* Net work done upon an object by an external
force  Changes the total mechanical energy (KE + PE)
of the object
 Positive work = object gained energy
 Negative work = object lost energy
Work- Energy Theorem
THE BIG CONCEPT!!
* Gain or loss in energy can be in the form of
 PE, KE, or both
Under such circumstances, the work which is done is
equal to the change in mechanical energy of the object
** External forces  capable of changing the total
mechanical energy of an object (Nonconservative forces)
Work-Energy Relationship
Analysis of situations involving external
forces
TMEi + WEXT = TMEf
(Initial amount of total mechanical energy (TMEi) plus the work done
by external forces (WEXT)  equals the final amount of total
mechanical energy (TMEf))
KEi + PEi + Wext = KEf + PEf
Practice Problems
DEFINITION OF A CONSERVATIVE FORCE
Version 1 A force is conservative when the work it
does on a moving object is independent of the path
between the object’s initial and final positions.
Version 2 A force is conservative when it does no
work on an object moving around a closed path,
starting and finishing at the same point.
Conservative Versus Nonconservative Forces
Conservative Versus Nonconservative Forces
Version 1 A force is conservative when the work it does on
a moving object is independent of the path between the
object’s initial and final positions.
Wgravity  mg ho  h f 
Conservative Versus Nonconservative Forces
Version 2 A force is conservative when it does no work
on an object moving around a closed path, starting and
finishing at the same point.
Wgravity  mg ho  h f 
ho  h f
Conservative Versus Nonconservative Forces
An example of a nonconservative force is the kinetic
frictional force.
W  F cos d  f k cos180 d   f k d
The work done by the kinetic frictional force is always
negative. Thus, it is impossible for the work it does on an
object that moves around a closed path to be zero.
The concept of potential energy is not defined for a
nonconservative force.
Conservative Versus Nonconservative Forces
In normal situations both conservative and nonconservative
forces act simultaneously on an object, so the work done by
the net external force can be written as
W  Wc  Wnc
W  KEf  KEo  KE
Wc  Wgravity  mgho  mgh f  PE o  PE f  PE
Conservative Versus Nonconservative Forces
W  Wc  Wnc
KE  PE  Wnc
THE WORK-ENERGY THEOREM
Wnc  KE  PE
The Conservation of Mechanical Energy
Wnc  KE  PE  KEf  KEo   PE f  PE o 
Wnc  KEf  PE f   KEo  PEo 
Wnc  Ef  Eo
If the net work on an object by nonconservative forces
is zero, then its energy does not change:
Ef  Eo
The Conservation of Mechanical Energy
THE PRINCIPLE OF CONSERVATION OF
MECHANICAL ENERGY
The total mechanical energy (E = KE + PE) of an object
remains constant as the object moves, provided that the net
work done by external nonconservative forces is zero.
The Conservation of Mechanical Energy
The Conservation of Mechanical Energy
Example A Daredevil Motorcyclist
A motorcyclist is trying to leap across the canyon by driving
horizontally off a cliff at 38.0 m/s. Ignoring air resistance, find
the speed with which the cycle strikes the ground on the other
side.
The Conservation of Mechanical Energy
Ef  Eo
mghf  mv  mgho  mv
1
2
2
f
1
2
gh f  12 v 2f  gho  12 vo2
2
o
The Conservation of Mechanical Energy
gh f  12 v 2f  gho  12 vo2
vf 


2 g h
o
 hf
 v
2
o
v f  2 9.8 m s 35.0m  38.0 m s   46.2 m s
2
2
Nonconservative Forces and the Work-Energy Theorem
THE WORK-ENERGY THEOREM
Wnc  Ef  Eo

 
Wnc  mghf  12 mv2f  mgho  12 mvo2

Nonconservative Forces and the Work-Energy Theorem
Example Fireworks
Assuming that the nonconservative force
generated by the burning propellant does
425 J of work, what is the final speed
of the rocket. Ignore air resistance. The mass
of the rocket is 0.2kg.


Wnc  mgh f  12 mv2f 
mgh
2
1

mv
o
o
2

Nonconservative Forces and the Work-Energy Theorem
Wnc  mghf  mgho  mv  mv
1
2
2
f
1
2
2
o
Wnc  mgh f  ho   12 mv2f


425 J  0.20 kg  9.80 m s 2 29.0 m 
 12 0.20 kg v 2f
v f  61m s
POWER
POWER:
* A quantity that measures the rate at which
work is done or energy is transformed
* Power = work / time interval
P = W/Δt
W = Fd P = Fd/Δt
* Power = Force x speed
P = Fv
 v = d/Δt
POWER
SI Unit for Power:
Watt (W)  Defined as 1 joule per second (J/s)
Horsepower = Another unit of power
1 hp = 746 watts
POWER PROBLEM
A 193 kg curtain needs to be raised 7.5 m,
in as close to 5 s as possible. The power
ratings for three motors are listed as 1
kW, 3.5 kW, and 5.5 kW. What motor is
best for the job?
POWER PROBLEM
m = 193 kg
Δt = 5s
d =7.5m
P=?
P = W/Δt
= Fd/Δt
= mgd/Δt
= (193kg)(9.8m/s2)(7.5m)/5s
= 280 W  2.8 kW
** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the
curtain fast enough, and the 5.5 kW motor will lift the curtain too fast
THE PRINCIPLE OF CONSERVATION OF ENERGY
Energy can neither be created nor destroyed, but can
only be converted from one form to another.
* Disclaimer: This powerpoint presentation is a compilation of various works.
Question
A cart is loaded with a brick and pulled at
constant speed along an inclined plane to the
height of a seat-top. If the mass of the loaded
cart is 3.0 kg and the height of the seat top is
0.45 meters, then what is the potential energy
of the loaded cart at the height of the seat-top?
PE = m*g*h
PE = (3 kg ) * (9.8 m/s/s) * (0.45m)
PE = 13.2 J
Question
If a force of 14.7 N is used to drag the
loaded cart (from previous question) along
the incline for a distance of 0.90 meters,
then how much work is done on the loaded
cart?
W = F * d * cos Theta
W = 14.7 N * 0.9 m * cos (0 degrees)
W = 13.2 J
(Note: The angle between F and d is 0 degrees because the
F and d are in the same direction)