Halliday, Resnick & Walker: Chapter 7-8
Kinetic Energy, Work,
Potential Energy,
Conservation of Energy
Physics 1A – PHYS1121
Professor Michael Burton
7-1 Kinetic Energy

Energy is required for any sort of motion

Energy:
o
A scalar quantity assigned to an object or system
o
Can be changed from one form to another
o

Is conserved in a closed system, that is the total amount of
energy of all types is always the same
Kinetic Energy is the energy associated with motion
7-1 Kinetic Energy


Kinetic Energy:
o
The faster an object moves, the greater its kinetic energy
o
Kinetic energy is zero for a stationary object
For an object with v well below the speed of light:
Eq. (7-1)

The unit of kinetic energy is a joule (J)
Eq. (7-2)
7-2 Work and Kinetic Energy


We account for changes in kinetic energy by saying
energy has been transferred to or from the object
In a transfer of energy via a force, Work is:
o

Done on the object by the force
This is not the common meaning of the word “work”!
o
To do work on an object, energy must be transferred
o
Throwing a cricket ball does work
o
Pushing an immovable wall does not do work
7-2 Work and Kinetic Energy

Start from force equation and 1-dimensional velocity:
Eq. (7-3)

Eq. (7-4)
Rearrange into kinetic energies:
Eq. (7-5)

The left side is now the change in kinetic energy

Therefore the work done is:
Eq. (7-6)
WEM02AN1: Introduction to Work
73 seconds
7-2 Work and Kinetic Energy


F
Φ
d
For an angle Φ between the force and the displacement:
As vectors we can write:
Eq. (7-7)
Eq. (7-8)

Notes on these equations:
o
Force must be constant here
o
Object is particle-like (rigid)
o
Work can be positive or negative
7-2 Work and Kinetic Energy


For two or more forces, the Net Work is the sum of the
contributions from all the individual forces
Two methods to calculate net work:


Sum the individual work terms for each force, or
Take the vector sum of forces (Fnet)
7-2 Work and Kinetic Energy

The work-kinetic energy theorem states:
Eq. (7-10)

i.e. change in kinetic energy = the net work done

Or we can write it as:
Eq. (7-11)

i.e. final KE = initial KE + net work
WEM04VD1: Work Done and Change in Speed
48 seconds
7-2 Work and Kinetic Energy

Work Done = Force x Distance = Increase in KE
Fx cos φ = 1 2 mv 2 − 1 2 mv 02

Work has the SI unit of joules (J), the same as energy
Ignoring friction effects, the amount of energy required to accelerate a car from rest
to a speed v is E. The energy is delivered to the car by burning petrol. What
additional amount of energy is required to accelerate the car to a speed 2v?
a) 0.5E
b) E
c) 2E
v, 2v
d) 3E
e) 4E
Work Done by Gravitational Force

Calculate the work as for any force
Φ

For a rising object:

For a falling object:

g
€
Particle thrown upward
7-3 Work Done by the Gravitational Force

Work done lifting object, applying upwards force:
“a” for “applied”


For a stationary object (e.g. lifting a book to shelf):
o
Kinetic energies are zero
o
So:
Thus, for an applied lifting force:
7-3 Work Done by the Gravitational Force


Figure shows orientations of
Forces and their associated
Work for upward and downward
Displacement
In general, we need to know the
initial and final Kinetic Energy to
solve for the Work
Figure 7-7
7-3 Work Done by the Gravitational Force
Work done by Normal Force :
W N = FN d cos90° = 0
Work done by Gravitational Force, Fg = mg, is :
W g = Fg sin θ ⋅ d⋅ cos180° = −Fg sin θ ⋅ d
Work done by Tension in rope is given by :
W N + W g + WT = ΔK = 0 if at rest at start & end
€
Thus WT = –Wg
In this example
Tension does positive work,
Gravity does negative work
7-3 Work Done by the Gravitational Force
Examples You are a passenger:
o
Being lowered down in an elevator
•  Tension does negative work,
Gravity does positive work
N2L
: Fg − T = ma
Gravity
: Fg = mg
Work Done : WT = Td cos φ = m(g − a)d cos φ
with φ = 180° in this case.
7-4 Work Done by a Spring Force



A spring force is a variable force
Fig. (a) shows the spring in its
relaxed state: since it is neither
compressed nor extended, no
force is applied.
If we stretch (b) or compress (c)
the spring it resists, and exerts
a restoring force that attempts
to return the spring to its relaxed
state.
Figure 7-10
7-4 Work Done by a Spring Force

The Spring Force is given by Hooke's law:
Eq. (7-20)




The negative sign means the force always opposes
the displacement
The spring constant k is a measure of the stiffness of
the spring
This is a variable force (a function of position). It
exhibits a linear relationship between F and d.
For a spring along the x-axis we can write:
Eq. (7-21)
7-4 Work Done by a Spring Force

We find the work by integrating:
xf
WS =


Substitute -kx for Fx:
The work:
∫ F dx
x
Eq. (7-23)
xi
€
o
Can be positive or negative
o
Depends on the net energy transfer
Eq. (7-25)
A block is in contact with a rough surface. The block has a rope attached to one side.
You pull the rope with a force as shown. The force is directed at angle θ with respect
to the horizontal. Its magnitude is equal to two times the magnitude of the frictional
force, f. For what value of θ is the net work on the block equal to zero joules?
a) 0°
2f
b) 30°
c) 45°
d) 60°
e) Net work will be done in the object for all values of θ.
f
7-5 Work Done by a General Variable Force



Take a 1D example
Need to integrate the Work
equation over the change
in position
Can be approximated with
rectangles under the curve
xf
WS =
∫ F dx
x
xi
W = lim ∑ F j,avg Δx
Δx → 0
7-6 Power

Power is the time rate at which a Force does Work

If a Force does Work W in a time Δt
the Average Power due to the Force is:

The Instantaneous Power at a particular time is:

The SI unit for Power is the watt (W): 1 W = 1 J/s

Therefore work-energy can be written as:
(power) x (time) e.g. kWh, the kilowatt-hour
Potential Energy
8-1 Potential Energy


Potential energy U is energy that can be associated
with the configuration of a system of objects that exert
forces on one another
A system of objects may be:
o
o
o

Earth and a bungee jumper
Gravitational potential energy accounts for kinetic energy
increase during the fall
Elastic potential energy accounts for deceleration by the
bungee cord
Physics determines how potential energy is calculated,
to account for the stored energy
8-1 Potential Energy, U

For an object being raised or lowered:
Eq. (8-1)


The change in gravitational potential energy is the
negative of the work done
This also applies to an elastic block-spring system
Figure 8-2
Figure 8-3
8-1 Potential Energy

Key points:
1.  The system consists of two or more objects
2.  A force acts between an object and rest of the system
3.  When the configuration changes, the force does work
W1, changing kinetic energy to another form
4.  When this is reversed, the force reverses the energy
transfer, doing work W2

Thus the kinetic energy of the object is converted to
potential energy, and then back to kinetic energy, etc.
W1
8-1 Potential Energy
W2


Conservative forces are forces where W1 = -W2
whatever paths are travelled from a to b and back to a.
o
Examples: gravitational force, spring force
o
Otherwise we could not speak of their potential energies
Nonconservative forces are those for which this is false
o
Examples: kinetic friction force, drag force
o
Kinetic energy of a moving particle is transferred to heat by friction
o
o
Thermal energy cannot be recovered back into kinetic energy of
the object via the friction force
Therefore the force is not conservative, thermal energy is not a
potential energy
8-1 Potential Energy


When only conservative forces act on a particle, we
find that many problems can be simplified:
A result of this is that:
Figure 8-4
8-1 Potential Energy


Mathematically:
Eq. (8-2)
This result allows you to substitute a simpler path for a
more complex one if only conservative forces are
involved
Path 1
Path 2
8-1 Potential Energy
a)  Yes, F is a conservative force.
b)  No, F is not conservative.
8-1 Potential Energy

For the general case, we calculate work as:
Eq. (8-5)

So we calculate potential energy as:
Eq. (8-6)

For F=-mg then the Gravitational PE at height y above
a reference point yi = 0 is:
Eq. (8-9)
8-1 Potential Energy

To calculate the PE of a spring:
Eq. (8-10)

With reference point xi = 0 for a relaxed spring:
Eq. (8-11)
U = −W
8-1 Potential Energy
€
a) Largest change in PE – (1), (2) or (3)?
b) Middle change in PE – (1), (2) or (3)?
c)  Smallest change in PE – (1), (2) or (3)?
WEM05VD2: Conversion of PE into KE
8-2 Conservation of Energy

The total energy of a system is the sum of its potential
energy U and kinetic energy K:
E = K +U

Eq. (8-12)
Work done by conservative forces that increases K
decreases U by the same amount, so:
Eq. (8-15)

€ to refer to different instants of time:
Using subscripts
Eq. (8-17)

In other words:
total
E
.
8-2 Conservation of Energy

This is the principle of Conservation of Energy:
ΔE = ΔK + ΔU = 0

This is very powerful tool:
€

One application:
o
Choose the lowest point in the system as U = 0
o
Then at the highest point U = max, and K = min
Eq. (8-18)
8-3 Reading a Potential Energy Curve

(a) is Potential U(x)

(b) is Force F(x)=-dU/dx


Draw a horizontal line for
the Total Energy, E=U+K
In (c), (d) & (e) x < x1 is
forbidden: the particle
does not have the energy
to reach these points
8-3 Reading a Potential Energy Curve

A particle in unstable equilibrium is stationary, with
potential energy only, and net force = 0
o
o

If displaced slightly to one direction, it will feel a force
propelling it in that direction
Example: a marble balanced on a bowling ball
A particle in stable equilibrium is stationary, with
potential energy only, and net force = 0
o
o
If displaced to one side slightly, it will feel a force returning it
to its original position
Example: a marble placed at the bottom of a bowl
The graph shows the potential energy as a function of distance for an object
moving along the x axis. At which of the labeled points does the force acting
on the object have the largest magnitude?
a) A
b) B
c) C
d) D
e) E
The graph shows the potential energy as a function of distance for an object
moving along the x axis. At which of the labeled points does the force acting on
the object have the least magnitude?
a) A
b) B
c) C
d) D
e) The force is the same at each of the four points.
The graph shows the potential energy as a function of distance for an object
moving along the x axis. At which of the labeled points does the object have
greatest speed?
a) A
b) B
c) C
d) D
e) E
8-4 Work Done on a System by an External Force


For a system of more than 1 particle, work can change
both K and U, or other forms of energy of the system
Consider lifting a ball in a frictionless system:
Eq. (8-25)
So W = ΔE
Figure 8-12
€
Eq. (8-26)
8-4 Work Done on a System by an External Force

For a system with friction force fk:
Eq. (8-31)
Fd = W = ΔE mec + ΔE th

Eq. (8-33)
The thermal energy (“heat”) comes from the forming
and breaking of the welds between the sliding surfaces
€
Figure 8-13
8-5 Conservation of Energy



Energy transferred between systems can always be
accounted for
The Law of Conservation of Energy concerns
o
The total energy E of a system
o
This includes mechanical, thermal, and other internal energy
Considering only energy transfer through work:
Eq. (8-35)
8-5 Conservation of Energy

An isolated system is one for which there can be no
external energy transfer

Energy transfers may happen internal to the system

We can write:
Eq. (8-36)
8-5 Conservation of Energy

External forces can act on a system without doing work:

The skater pushes herself away from the wall

Turns internal chemical energy in muscles into KE, K

The change in K is caused by the force from the wall,
but the wall does not provide the energy
(ΔU+) ΔK = Fd cos φ
8-5 Conservation of Energy
• Engine causes the tyres to push backward on the road.
• Produces frictional forces which push forward, moving the car.
• Net external force comes from the road, but it does not transfer energy to the
car, and so does no work. The KE is increased due to the transfer of chemical
energy from the combustion of petrol.
8-5 Conservation of Energy


Now we expand our definition of Power.
Power is the rate at which energy is transferred by a force
from one type to another

If energy ΔE transferred in time Δt, the average power is:

And the instantaneous power is:
WEA09AN1: Gravitational PE and Escape Speed
7
Summary
Kinetic Energy

Work
The energy associated with
motion
Eq. (7-1)
Work Done by a Constant
Force
Eq. (7-7)
Eq. (7-8)

The net work is the sum of the
individual contributions


Energy transferred to or from an
object via a force
Can be positive or negative
Work and Kinetic Energy
Eq. (7-10)
Eq. (7-11)
7
Summary
Work Done by the
Gravitational Force
Work Done in Lifting and
Lowering an Object
Eq. (7-12)
Eq. (7-16)
Spring Force


Spring Force
Relaxed state: applies no force
Spring constant k measures
stiffness
Eq. (7-20)

For an initial position x = 0:
Eq. (7-26)
7
Summary
Work Done by a Variable
Force

Found by integrating the
constant-force work equation
Power


The rate at which a force does
work on an object
Average power:
Eq. (7-42)
Eq. (7-32)

Instantaneous power:
Eq. (7-43)

For a force acting on a moving
object:
Eq. (7-47)
Eq. (7-48)
8
Summary
Conservative Forces

Net work on a particle over a
closed path is 0
Potential Energy

Energy associated with the
configuration of a system and a
conservative force
Eq. (8-6)
Gravitational Potential
Energy

Energy associated with Earth +
a nearby particle
Eq. (8-9)
Elastic Potential Energy

Energy associated with
compression or extension of a
spring
Eq. (8-11)
8
Summary
Mechanical Energy
Potential Energy Curves
Eq. (8-22)
Eq. (8-12)

For only conservative forces
within an isolated system,
mechanical energy is
conserved
Work Done on a System by
an External Force

Without/with friction:
Eq. (8-26)
Eq. (8-33)


At turning points a particle
reverses direction
At equilibrium, slope of U(x) is 0
Conservation of Energy

The total energy can change
only by amounts transferred in
or out of the system
Eq. (8-35)
8
Summary
Power


The rate at which a force
transfers energy
Average power:
Eq. (8-40)

Instantaneous power:
Eq. (8-41)