Transcript potential energy

Lecture 7, Potential Energy,
Conservation of Energy
Dale Gary
Recall Last Time
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Learned that work is force through a distance W   F  ds
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For a constant force, can take it out of the integral W  F   ds  Fd cos q
Here, d is the distance over which the (constant) force acts, and q is the
angle between the force and the direction of movement.
Doing work on an object gives it kinetic energy
W  K f  Ki  12 mv f  12 mvi
2
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Non-Constant Force—Springs
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W
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There are times when the force is not
constant.
An example is a spring, for which the
force depends on the distance the
spring is stretched or compressed.
The force can be written F  kx
where k is the spring constant. Why
negative? Because the force is in the
opposite direction to the
displacement.
Now, since the force is not constant,
calculating the work is trickier:
 
f
  F  ds    kxdx  k  xdx  12 kxi2  12 kx2f
i
It is the area under the force curve.
Gravitational Potential Energy
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When you drop an object, it accelerates toward the ground and may be
moving rather fast by the time it hits. Therefore, it has a lot of kinetic
energy.
Where did that energy come from?
We can consider that it had the energy by virtue of the fact that it was held
at some height above the ground. This is potential energy. It is stored in
the object, waiting to be released.
An object sitting on the ground, however, has no potential energy. We give
it potential energy by doing work on it, i.e. by lifting it to some higher height,
against the force of gravity. Imagine an applied force Fapp = mg that
balances the force of gravity, and we lift the object from height yi to yf. We
then do work
 
f
W   F  ds   mgdy  mg  dy  mgy f  mgyi
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But the object is not moving at the end of our lift, so the energy is potential
energy, stored in the object by virtue of its location.
We use the symbol U for potential energy, and identify U g  mgy
(assumes yi = 0)
Gravitational potential energy
Spring Potential Energy
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Gravitational potential energy is not the only
kind.
A spring has potential energy by virtue of the
compression or stretching of the spring.
We just saw that the work done by the spring
is
W  1 kx2  1 kx2
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f
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The work done by an applied force
compressing the spring is just the negative of
this, and it is this work that is stored in the
object, i.e. the spring potential energy
U s  W  12 kx2f  12 kxi2
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In short, assuming xi = 0, the spring potential
energy is
U s  12 kx2
Spring potential energy
Relationship Between K and U
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Placing a system in a certain configuration (like compressing or stretching a
spring, or lifting an object to some height) can store energy.
When the system is “released,” that energy is available to do work on the
object, i.e. make it move and give it energy of motion (kinetic energy).
By the same token, a moving object, with kinetic energy, can move into a
configuration in which the energy of motion is stored in the object, as when
a ball is thrown upward. The ball moves upward and eventually comes to a
stop. At that moment, high above the ground, the kinetic energy is gone,
and is converted into potential energy.
Here is a similar example with a spring.
Animation
Conservative Forces
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Both gravity and a spring are examples of conservative forces.
There are two properties that conservative forces have.
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2.
The work done by a conservative force on a particle moving between any two
points is independent of the path taken by the particle.
The work done by a conservative force on a particle moving through a closed
path is zero.
B
Work done in going from A to B
is the same for any and all paths
A
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Non-Conservative Forces
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Not all forces are conservative. In fact, whenever friction is important in a
physical situation (i.e. nearly always), the friction spoils this neat setup.
Take the example we just gave.
If friction is acting along the ramps, then the longer ramp will have more
friction, so the ball will be moving more slowly after going along the longer
ramp.
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Getting the Force from the Energy
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We have the work done by the component of a conservative force in the x
xf
direction is
W   Fx dx   U
xi
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So the change in potential energy is
xf
U  U f  U i    Fx dx
xi
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An infinitesimal change in potential energy results from an infinitesimal
increment dx, i.e. dU   Fx dx
dU
So we have the differential relationship Fx  
dx
Likewise in the other directions: Fy   dU
dy
In the case of gravity:
dU
dmgy
Fy  

 mg
dy
dy
Fz  
dU
dz
In the case of a spring:
d 12 kx2
dU
Fx  

 kx
dx
dx
The Potential Energy Function
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For a spring, the potential energy
function looks like this:
Then the force is the negative of the
slope of this function.
This kind of curve is called a
potential well (looks like a bowl), and
it indicates a stable equilibrium.
Animation
Unstable Equilibrium
Conservation of Energy
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One of the great laws of physics is the conservation of energy.
What this law says is that energy can be neither created nor destroyed. It
can only be changed from one form to another.
A corollary to this law is that for an isolated system the energy is constant.
What do we mean by an isolated system? It means the system does not
interact with anything outside the system. For example, a falling apple,
together with the Earth, can make an isolated system (but we have to ignore
the Earth’s interaction with the Sun, the Moon, etc).
Of course, no system is truly isolated. In a non-isolated system, the energy
does not have to be constant. Energy can come in from outside, or go out
from within. An example is mass and a spring, but with friction. Now
energy can leave the “system” through friction with the table (which takes
the form of heat). We can account for that if we add a term Eint.
Conservation of energy states that K  U  Eint  0
where we have ignored other loss of energy due to transfer out of the
system.
Case of Conservative Forces
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When we ignore friction or other non-conservative forces, we have
K  U  ( K f  K i )  (U f  U i )  0 or K f U f  K i U i
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Example, dropping a ball from rest. Initial Ki=0,
and initial height is h. So total energy is just mgh.
While falling, the potential energy at position y is
U = mgy, and kinetic energy is ½mv2, so the total
energy is just the sum of these.
At the floor, y = 0, so Uf = 0 and Kf = ½mvf2.
The final energy equals the initial energy, so
K f U f  12 mv2f  Ki U i  mgh
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And we can solve for vf to get
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This is the same result we got when using forces.
v f  2 gh
Case of Friction (non-conservative)
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In many cases, the friction force is constant (remember that kinetic friction
depends on the normal force, which is just the weight of an object in the
simplest case).
Consider a block sliding on a rough surface. If it slides a distance d = x,
the force of friction fk through that distance gives the form of energy we
called internal energy (heat): Eint  f k d
Since
K  U  Eint  0
then
K f U f  K i U i  f k d
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For this problem, Ui = Uf = 0, so you
can see that the final kinetic energy
will be lower than the non-friction
case.
Example, crate sliding down ramp
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Initial energy is mgy.
Crate slides 1 m against friction
force, so Eint = fkd = 5 Nm.
Final energy is all kinetic,
½ mvf2.
Putting it all together,
Find vf.
K f U f  K i U i  f k d
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mv2f  mgy  f k d
v f  2 gy 
2 fk d
m
v f  2(9.8m/s 2 )0.5m 
2(5 N)1m
 2.54 m/s
3kg
fk = 5 N
m = 3 kg
Power
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A useful concept is the amount of energy used per unit time, which is called
dW
power:
P
dt
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Now, since W   F  ds , we have dW  F .ds
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dW  ds  
But we can write P 
F
 F v
dt
dt
which assumes that F is constant.
The units of power are watts, 1 W = 1 J/s