Download Chapter 7 - Potential Energy & Energy Conservation

Chapter 7
Potential Energy and
Energy Conservation
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
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Goals for Chapter 7
• To use gravitational potential energy for vertical
motion
• To use elastic potential energy for a body
attached to a spring
• To solve problems involving conservative and
non-conservative forces
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8-1 Potential Energy

Potential energy (symbol = U) is energy (Joules!)


Associated with 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 (KE increases!)
Elastic potential energy accounts for
deceleration by the bungee cord
(KE decreases)
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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
Configuration means that
WHERE objects are will matter.

Physics determines how
potential energy is
calculated
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8-1 Potential Energy

But note: Energy (in the universe) is conserved!



Energy can be transformed from potential to
kinetic…
Energy can be transformed from kinetic to
potential….
Energy can be transformed into thermal
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Potential Energy
• Energy associated with a particular position of
a body when subjected to or acted on by forces.
–Field Forces act on bodies even if not
touching, like gravity, magnetism,
electricity
–Direct Contact forces, like springs
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Gravitational potential energy
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Gravitational potential energy
When a particle is in the gravitational field of the
earth, there is a gravitational potential energy
associated with the particle:
• As the basketball descends,
gravitational potential
energy is converted to
kinetic energy and the
basketball’s speed
increases.
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Potential Energy
• Energy associated with position is potential
energy.
– Example: Gravitational potential energy is
Ugrav = mgy for a position “y”.
• But what is “y”? Where is “y” = 0??
• Potential Energy is RELATIVE, not absolute…
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Gravitational potential energy
When a particle is in the gravitational field of the
earth, there is a gravitational potential energy
associated with the particle:
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Gravitational Potential Energy


For any object being raised or lowered:
The change in gravitational potential energy is the
negative of the work done by the force of gravity
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Moving Downwards…
• Change in
gravitational
potential energy
is related to
work done by
gravity.
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Moving Downwards
• Change in gravitational potential energy is related to
work done by gravity.
Work done by Gravity:
• Force ● Distance
• Force = mg (down)
• Distance = |Dy| (down)
• Angle between: 0
• Work done = POSITIVE
Work = +mg Dy
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Moving Downwards….
• Gravitational potential energy is related to configuration
of objects (mass “m” and Earth)
Define potential energy of a
position at height “h”
relative to “0” as
mgh
NOTE – where “0” is will be
YOUR choice…
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Gravitational potential energy
• Change in gravitational potential energy is related to
work done by gravity.
Work done by Gravity:
+mgDy
Initial Potential Energy:
mgy1 (higher)
Final Potential Energy:
mgy2 (lower!)
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Moving DOWN
• Change in gravitational potential energy is related to
work done by gravity.
Work done by Gravity:
+mgDy (positive!)
Initial Potential Energy:
mgy1 (start high!)
Final PE:
mgy2
(lower!)
Difference: PEfinal – PE initial
(mgy2 – mgy1) < 0
Negative!
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Gravitational potential energy
• Change in gravitational potential energy is related to
work done by gravity..
OK… what about
moving
UPWARDS??
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Gravitational potential energy
• Change in gravitational potential energy is related to
work done by gravity..
Work done by Gravity:
- mgDy
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Gravitational potential energy
• Change in gravitational potential energy is related to
work done by gravity..
Work done by Gravity
(up!):
- mgDy
(negative!)
Initial Potential Energy:
mgy1 (lower)
Final PE:
mgy2 (higher!)
Difference: PEfinal – PE initial
(mgy2 – mgy1) > 0
Positive!
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Gravitational potential energy
• Either moving DOWN or UP, change in gravitational
potential energy is equal in magnitude and opposite in sign
to work done by gravity.
Work done by Gravity up:
- mgDy
Difference: PEfinal – PE initial
DU = (mgy2 – mgy1) > 0
DU = positive!
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Gravitational potential energy
• Either moving DOWN or UP, change in gravitational
potential energy is equal in magnitude and opposite in sign
to work done by gravity.
Work done by Gravity down:
+ mgDy
Difference: PEfinal – PE initial
DU = (mgy2 – mgy1) < 0
DU = negative!
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
For an object being raised or lowered:
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The conservation of mechanical energy
• The total mechanical energy of a system is the
sum of its kinetic energy and potential energy.
• A quantity that always has the same value is
called a conserved quantity.
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The conservation of mechanical energy
• When only force of gravity does work on a system,
total mechanical energy of that system is conserved.
• This is an example of the conservation of mechanical
energy.
• Gravity is known as a “conservative” force
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An example using energy conservation
• 0.145 kg baseball thrown straight up @ 20m/s. How high?
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An example using energy conservation
• 0.145 kg baseball thrown straight up @ 20m/s. How high?
• Use Energy Bar Graphs to track total, KE, and PE:
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When forces other than gravity do work
• Now add the
launch force!
(move hand
.50 m upward
while
accelerating
the ball)
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When forces other than gravity do work
• Now add the
launch force!
(move hand
.50 m upward
while
accelerating
the ball)
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Work and energy along a curved path
• We can use the same
expression for
gravitational
potential energy
whether the body’s
path is curved or
straight.
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Energy in projectile motion – example 7.3
• Two identical balls leave from the same height with the
same speed but at different angles. Prove they have the
same speed at any height h (neglecting air resistance)
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Motion in a vertical circle with no friction
• Speed at bottom of ramp of radius R = 3.00 m?
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Motion in a vertical circle with no friction
• Speed at bottom of ramp of radius R = 3.00 m?
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Motion in a vertical circle with no friction
• Normal force DOES NO WORK!
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Motion in a vertical circle with friction
• Revisit the same ramp as in the previous example, but this time
with friction.
• If his speed at bottom is 6.00 m/s, what was work by friction?
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Moving a crate on an inclined plane with friction
• Slide 12 kg crate up 2.5
m incline without
friction at 5.0 m/s.
• With friction, it goes
only 1.6 m up the slope.
• What is fk?
• How fast is it moving at
the bottom?
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Moving a crate on an inclined plane with friction
• Slide 12 kg crate up 2.5
m incline without
friction at 5.0 m/s.
• With friction, it goes
only 1.6 m up the slope.
• What is fk?
• How fast is it moving at
the bottom?
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Work done by a spring
• Work on a block as spring is stretched and compressed.
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Elastic potential energy
• A body is elastic if it returns
to its original shape after
being deformed.
• Elastic potential energy is
the energy stored in an
elastic body, such as a
spring.
• Figure shows a graph of the
elastic potential energy for
an ideal spring.
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Elastic potential energy
The elastic potential energy stored in an ideal spring is
Uel = 1/2 kx2.
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Situations with both gravitational and elastic forces
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Situations with both gravitational and elastic forces
I can calcuate
exactly how high I’ll
go using
conservation of
energy!
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Situations with both gravitational and elastic forces
But can you
calculate the force
on each paw as you
land??
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Situations with both gravitational and elastic forces
• When a situation involves
both gravitational and
elastic forces, the total
potential energy is the sum
of the gravitational potential
energy and the elastic
potential energy:
U = Ugrav + Uel.
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Motion with elastic potential energy
• Glider of mass 200 g on frictionless air track, connected to spring
with k = 5.00 N/m. Stretch it 10 cm, and release from rest.
• What is velocity when x = 0.08 m?
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Motion with elastic potential energy
• Glider of mass 200 g on frictionless air track, connected to spring
with k = 5.00 N/m. Stretch it 10 cm, and release from rest.
• What is velocity when x = 0.08 m?
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A system having two potential energies and friction
• Gravity, a spring,
and friction all act on
the elevator.
• 2000 kg elevator
with broken cables
moving at 4.00 m/s
• Contacts spring at
bottom, compressing
it 2.00 m.
• Safety clamp applies
constant 17,000 N
friction force as it
falls.
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A system having two potential energies and friction
• What is the spring
constant k for the
spring so it stops in
2.00 meters?
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Conservative and nonconservative forces
• A conservative force allows conversion between kinetic and
potential energy. Gravity and the spring force are
conservative.
• The work done between two points by any conservative force
a) can be expressed in terms of a potential energy function.
b) is reversible.
c) is independent of the path between the two points.
d) is zero if the starting and ending points are the same.
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Conservative forces
The work done by
a conservative
force such as
gravity depends
ONLY on the
endpoints of a
path, not the
specific path
taken between
those points.
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
Mathematically:
This result allows you to substitute a simpler path for a
more complex one if only conservative forces are
involved
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Answer: No. The paths from a → b have different signs. One pair of
paths allows the formation of a zero-work loop. The other does not.
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8-1 Potential Energy
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8-1 Potential Energy
Answer: (3), (1), (2); a positive force does positive work, decreasing the
PE; a negative force (e.g., 3) does negative work, increasing the PE
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Conservative and nonconservative forces
• A conservative force allows conversion between kinetic and
potential energy. Gravity and the spring force are
conservative.
• A force (such as friction) that is not conservative is called a
non-conservative force, or a dissipative force.
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Frictional work depends on the path
• Move 40.0 kg futon 2.50 m across room; slide it
along paths shown. How much work required if
mk = .200
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Nonconservative forces
As an automobile tire flexes as it rolls, nonconservative
internal friction forces act within the rubber.
Mechanical energy is lost and converted to internal energy of
the tire.
This causes the temperature
and pressure of a tire to
increase as it rolls.
That’s why tire pressure is best
checked before the car is
driven, when the tire is cold.
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Conservation of energy
Nonconservative forces do not store potential
energy, but they do change the internal energy of a
system.
The law of conservation of energy means that
energy is never created or destroyed; it only
changes form.
This law can be expressed as
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Conservative or nonconservative force?
• Suppose force F = Cx in the y direction. What is work
required in a round trip around square of length L?
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Force and potential energy in one dimension
In one dimension, a conservative force F(x) can
be obtained from its potential energy function
U(x) using by looking at the rate of change…
.
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Force and potential energy in one dimension
In regions where U(x) changes most rapidly with
x, this corresponds to a large force magnitude.
Also, when Fx(x) is in the positive x-direction, U(x)
decreases with increasing x.
A conservative force always acts to push the
system toward lower potential energy.
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Force and potential energy
The greater the elevation of a hiker in Canada’s
Banff National Park, the greater the
gravitational potential energy Ugrav.
.
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Force and potential energy
Where the mountains
have steep slopes,
Ugrav has a large
gradient
There’s a strong force
pushing you along
mountain’s surface
toward a region of
lower elevation
(and lower Ugrav).
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Force and a graph of its potential-energy function
• For any graph of potential energy versus x, the
corresponding force is Fx = −dU/dx.
• Whenever the slope of U is zero, the force
there is zero, and this is a point of equilibrium.
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Force and potential energy in two dimensions
• In two dimension, the components of a conservative
force can be obtained from its potential energy
function using
Fx = –U/dx
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and
Fy = –U/dy
Consider a standard elevation plot:
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Where is the slope STEEPEST?
Most gentle?
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Since U(gravity) = mgh, lines of equal “h” are lines
of equal potential energy for any mass m
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The gradient is PERPENDICULAR to lines of equal
potential energy; where the gradient is largest =
steepest slope (greatest force!)
Greatest
change in
height in
smallest
horizontal
distance
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Force and a graph of its potential-energy function
• For any graph of potential energy versus x & y,
the corresponding forces are
• Fx = −dU/dx
• Fy = −dU/dy
• Whenever the slope of U is zero, the force
there is zero, and this is a point of equilibrium.
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Force and potential energy in three dimensions
In three dimensions, the components of a
conservative force can be obtained from its
potential energy function using partial
derivatives:
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Force and potential energy in three dimensions
When we take the partial derivative of U with
respect to each coordinate, multiply by the
corresponding unit vector, and then take the
vector sum, this is called the gradient of U:
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Unstable equilibrium
Each of these acrobats is in
unstable equilibrium.
The gravitational potential energy
is lower no matter which way
an acrobat tips, so if she
begins to fall she will keep on
falling.
Staying balanced requires the
acrobats’ constant attention.
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Force and a graph of its potential-energy function
• When U is at a minimum, the force near the
minimum draws the object closer to the
minimum, so it is a restoring force. This is
called stable equilibrium.
• When U is at a maximum, the force near the
maximum draws the object away from the
maximum. This is called unstable
equilibrium.
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Force and potential energy in one dimension
Elastic potential energy and force as functions of
x for an ideal spring.
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Force and potential energy in one dimension
Gravitational potential energy and the
gravitational force as functions of y.
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Force and a graph of its potential-energy function
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Force and a graph of its potential-energy function
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Force and a graph of its potential-energy function
Answer: (a) CD, AB, BC
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(b) to the right
Force and a graph of its potential-energy function

Plot (a) shows the potential U(x)

Plot (b) shows the force F(x)
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Force and a graph of its potential-energy function

If we draw a horizontal line, we can see the range of
possible positions
x < x1 is
forbidden for the
Emec in (c): the
particle does not
have the energy
to reach those
points
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Force and a graph of its potential-energy function
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Force and a graph of its potential-energy function
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Force and a graph of its potential-energy function
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Energy diagrams
An energy diagram is a
graph that shows both
the potential-energy
function U(x) and the
total mechanical
energy E.
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