Ch 8
Potential energy
and Conservation of Energy
Question: 2, 3, 8, 9
Problems: 3, 9, 15, 21, 24, 25, 31, 32,
35, 41, 43, 47, 49, 53, 55, 63
Potential energy
Kinetic energy – energy due to motion
Potential energy – energy due to the arrangement of the
objects in a system (U)
Potential energy is due to an objects location.
gravitational potential energy – due to the separation
between two objects
elastic potential energy – energy associated with the
compression or stretching of an elastic object
Potential energy is a form of energy so the units are Joules.
Potential energy is energy that can be stored and later
released to produce motion.
Examples:
Shove a ball into a spring loaded gun. The elastic potential
energy is stored and when you shoot the ball the potential
energy is converted to kinetic energy.
After climbing a tree you have gravitational potential
energy stored up. Fall out of the tree that potential
energy is converted to kinetic energy as you fall down.
Work and potential energy
Throw ball vertically upward.
As the ball rises, gravity does negative work on the ball.
The ball slowed down (lost kinetic energy).
The kinetic energy is transferred to the gravitational
potential energy of the ball-Earth system.
As the ball is falling, gravity does positive work on the ball.
The ball speeds up (gains kinetic energy).
The kinetic energy comes from the gravitational
potential energy of the ball-Earth system.
As ball rises:
K<0
When the gravitational force did negative work, the
GPE increased, U > 0.
As ball falls:
K>0
When the gravitational force did positive work, the
GPE decreased, U < 0.
From chapter 7: W = K
W=- U
Block hitting spring
Replace throwing a ball upward with a block sliding into a
spring that is fixed to a wall.
When the block hits the spring the block has kinetic energy.
As the block compresses the spring, it loses kinetic energy
and spring gets elastic potential energy.
When block comes to rest, all the kinetic energy has been
turned into potential energy.
As the spring expands the transfer of energy is reversed from
potential energy to kinetic energy.
Conservative forces
Assume a force acts between a system of 2 objects.
As the force changes the configuration of the system, the force
does work, W1, energy is transferred from kinetic energy to
some other form of energy.
When the change in configuration is reversed, the force does
work, W2, and the transfer of energy is reversed.
If W1 = -W2 is true the other type of energy was potential
energy.
The force is a conservative force.
examples: of conservative forces: gravity, springs, electric force
If W1 = -W2 is not always true, the force is nonconservative.
Examples of nonconservative forces: friction, drag
Nonconservative forces convert kinetic energy to thermal
energy. Transfers to thermal energy cannot be reversed.
Example: Kinetic energy can be lost to friction, but
friction cannot be used to produce kinetic energy.
Conservative forces are path independent.
The work done by a conservative force only depends on the
endpoint.
How you get from the initial configuration to the final
configuration does not matter.
The net work done by a conservative force around a closed
loop is zero.
Example: Throw a ball up and let it fall back down. The total
work done by gravity is zero.
Since conservative forces are path independent, we usually
only care about the endpoints. This can make a complicated
problem easier to solve.
Nonconservative forces are path dependent.
When dealing with nonconservative forces, such as friction, we
need to consider the path the object takes.
|Wf| = Ff d where d is the path length.
A
2
B
1
Drag a box over a rough surface from point 1 to point 2. Friction
will do more work when using path A than path B.
If you dragged the box along A from 1 to 2 then dragged it back
using path B the total work done by friction is not zero.
How to calculate values of potential energy.
In general, the work done by a force is:
W
xf
xi
F ( x)dx
Since U = -W
U
xf
xi
F ( x)dx
Gravitational Potential Energy
U
U
yf
yi
( mg )dy
mg y
mg
yf
yi
(picking up to be positive)
dy
mg ( y f
yi )
Gravitational Potential Energy
Change in GPE from one point to another is:
U
yf
yi
( mg )dy
mg
yf
yi
dy
mg ( y f
yi )
U mg y
Only changes in GPE are physically meaningful.
U – Ui = mg(y – yi)
We will often compare GPE to the GPE at a reference point.
Usually we use: Ui = 0 at yi = 0.
So: U(y) = mgy
You can set any height to be the zero of GPE. This will be a
big convenience in solving problems.
Elastic potential energy (EPE)
1 2 1 2
U
( kx)dx k xdx
kxf
kxi
xi
xi
2
2
When spring is relaxed, xi = 0, the Ui = 0
xf
xf
U – 0 = ½ kx2 – 0
U(x) = ½ kx2
Remember from last chapter Ws = - ½ kx2
So this fits U = -W
Conservation of mechanical energy
Mechanical energy: Emech = K + U
If you only have conservative forces, the total
mechanical energy never changes.
K2 + U2 = K1 + U1
and
Emech = K + U = 0
Emech = K + U = 0
This is definitely one of the most important
and most useful rules in physics.
If there are only conservative forces involved,
and there are multiple steps, you can ignore
the intermediate configurations of the system.
Will see a good example of this later.
Potential Energy Curves
Graph of potential energy vs. position
See fig. 8-9
For a particle moving in the x-direction
U(x) = -W = -F(x) x
F
U ( x)
x
dU ( x)
dx
The derivative (slope) of the potential energy
curve is related to the force.
The force doing the work, is the negative of
the slope in the potential energy curve.
Example: spring
U(x) ½ kx2
F ( x)
dU ( x)
dx
d 1 2
( kx )
dx 2
kx
U(x)
x
U(x) + K(x) = Emech
K(x) = Emech – U(x)
Where U(x) = Emech the kinetic energy will be zero.
This is a turning point.
Think of the potential energy curve as a rollercoaster track.
The height of the track is U(x). When the cart reach a
point where its total energy is U(x) , the cart stops and
turns around.
In Newtonian (classical physics), particles are confined
between turning points.
Equilibrium points.
Points on the potential energy curve are said
to be either unstable or stable.
A point is unstable is a small displacement in
either direction will lead to a greater
displacement. (relative maxima)
A point is stable is after a small displacement
in either direction, the particle will return to
its original position. (relative minima)
Work done by an external force.
If positive work is done by an external force on a
system, energy is transferred to the system.
Negative work done on a system results in
energy transferred from the system.
W = U + E = Emech
Friction
Look at case where a force, F, dragged a box
across a floor.
Newton’s 2nd law gives: F – fk = ma
F
ma
fk
a is constant so:
v 2 vo2 2ad
(v 2 v02 )
a
2d
1 2 1 2
Fd
mv
mvo
2
2
Fd
K fk d
fk d
The friction between the two surfaces causes
their temperatures to rise.
The sliding increases the thermal energy by an
amount equal in magnitude to the work done
by friction: Eth = fkd
Fd = Emech + fkd = Emech + Eth
Work done on a system when there is friction:
W = Emech + Eth
Conservation of Energy
The total energy of a system can change only
by amounts of energy that are transferred to or
from the system.
Energies to be considered are mechanic,
thermal, and internal
When work is done, the work is equal to the
sum of the changes in these types of energies.
W = E = Emech + Eth + Eint
Conservation of energy for an isolated system:
Emech + Eth + eint = 0
For an isolated system, the total energy cannot change.
Power:
Average power is the average rate that a force
transfers energy from, one type to another.
E
t
Pave
Instantaneous power:
P
dE
dt
Problems: 18, 26, 34, 44, 46, 58