Transcript Review - Canvas

Potential Energy Curves
Notes and Virtual Lab Activity – AP Mechanics
Energy and Work
Lab
Part 2
Click on the picture below to be directed to pHet’s virtual
skate-park lab. (Click “run now!” once on site.)
Use the lab handout to set the parameters for each portion of
this lab then use the virtual lab to investigate work and energy
and answer the lab questions.
Energy and Work
Potential
Energy
Curves
We already know that
x2
W 
 F ( x)dx  area
x1
We also know that work (done by a force) causes a change in energy.
Consider the following…
If we want to lift this bowling ball we have to
apply a force and WE have to do work to it. The
Fapp
work we do to the ball would be called the work applied
(because our applied force is acting through some distance).
h
mg
As the ball moves upward the work applied is
positive (increasing the potential energy) but the
work done by gravity is negative (because mg is
down but the motion is up).
Energy and Work
Potential
Energy
Curves
Let us assume that the ball was raised at a constant speed (a=0). We
know then that the magnitudes of Fapp and mg are equal.
In raising the ball the work applied is Wapp= Fapph = mgh.
This work (Wapp= mgh) increased the potential energy
so we write:
Fapp
Wapp= +∆PE= +∆U
mg
h
Energy and Work
Potential
Energy
Curves
Let us assume that the ball was raised at a constant speed (a=0). We
know then that the magnitudes of Fapp and mg are equal.
In raising the ball the work done by gravity is
Wg= -Fgh = -mgh.
Notice that the value (no sign included) is the same as the work applied.
Fapp
This work (Wg= -mgh) is opposing the increase in the
potential energy so we write:
mg
h
Wg= -∆PE= -∆U
Energy and Work
Potential
Energy
Curves
Let us look at this in another way.
What happens when we let go of the ball?
Surprise!
The ball falls.
Fapp
As it falls gravity does POSITIVE work on the ball and
the potential energy DECREASES.
mg
h
+Wg= -∆PE= -∆U
The FIELD will ALWAYS WORK to
REDUCE the POTENTIAL ENERGY!
Energy and Work
So now we know…
x2
W 
 F ( x)dx  area
Potential
Energy
Curves
Or focus is with a gravitational field,
but this is true for any type of f
ield OR restoring force.
+Wfield= -∆PE= -∆U
x1
x2

U   F ( x)dx  area
x1
You have to STOP and THINK about the relationship between the signs of W and
∆U!
If the force is causing an increase in the potential energy then both W and ∆U are positive. If the force is
causing a decrease in the potential energy then ∆U will be negative.
Energy and Work
Potential
Energy
Curves
x2

U   F ( x)dx  area
x1
If potential energy is the (negative) antiderivative of force (with respect to
displacement) then how would we find the force if we were given a potential
energy function?
Just go the opposite way….
…the reverse process of the antiderivative is the derivative.
 dU
F ( x) 
  slope
dx
Energy and Work
Potential
Energy
Curves
x2

U   F ( x)dx  area
x1
The area of a Force vs Position graph gives the work done by that force.
The opposite of the area of a force vs position graph give the change in potential energy.
The opposite of the slope of a potential energy vs position graph gives the force acting
on that particle.
 dU
F ( x) 
  slope
dx
Energy and Work

Potential
Energy
Curves
Potential Energy Curves graphically represent how the potential energy of
a moving particle changes with its position.
 Three “Flavors”




Stable Equilibrium
Unstable Equilibrium
Neutral Equilibrium
Equilibrium  occurs when the net force acting on an object is zero,
resulting in zero acceleration (Fnet = ma = 0).

Considering what we just learned, that means for a graph of potential energy vs
position (known as a potential energy curve), we want to
look for F ( x)  dU   slope  0 to identify points of equilibrium.
dx
Energy and Work
Stable Equilibrium – think back to the pHet Skater Lab.
Potential
Energy
Curves
Due to the starting position of the
skater, there was a certain total amount of energy available to the system.
E
Total Energy
U
0
x
As the skater moved,
her potential energy
increased and
decreased.
Energy and Work
Potential
Energy
Curves
Stable Equilibrium – occurs when a SMALL displacement in the particle results in a
restoring force that accelerates the particle back to the origin (its equilibrium
position).
Visualize the skater – a
small displacement to the
left (-x) would result in a
restoring force which is
positive (to the right).
This would return her to
the origin.
E
Total Energy
U
F ( x) 
F(x) = -dU/dx = -slope
Because the slope is
negative, the force is
positive.
0
F(x) = -dU/dx = 0
x
 dU
  slope
dx
When the skater is at x=0
the slope is zero; this
represents an equilibrium
point (which happens to
be stable).
Energy and Work
Potential
Energy
Curves
Unstable Equilibrium – occurs when a SMALL displacement in the particle results
in a restoring force that accelerates the particle AWAY FROM the origin (its
equilibrium position).
Visualize the skater –if he
stands atop a ramp that is
concave down and he is
displaced to the left, he
will not return to his
starting position.
E
Total Energy
U
0
If he was displaced (off of either side) his potential
energy would decrease.
x
He does, however, have
energy due to his position
Energy and Work
Potential
Energy
Curves
Unstable Equilibrium – occurs when a SMALL displacement in the particle results
in a restoring force that accelerates the particle AWAY FROM the origin (its
equilibrium position).
Visualize the skater – a
small displacement to the
left (-x) would result in a
force which is negative (to
the left). This would
accelerate him away from
the origin.
E
Total Energy
U
F(x) = -dU/dx = -slope
Because the slope is
positive, the force is
negative.
0
F(x) = -dU/dx = 0
F ( x) 
x
 dU
  slope
dx
When the skater is at x=0
the slope is zero; this
represents an equilibrium
point (which happens to be
unstable).
Energy and Work
Potential
Energy
Curves
Neutral Equilibrium – occurs when a SMALL displacement in the particle results in
no net force and the particle remains at rest.
E
Total Energy
U
0
x
Visualize the skater –if he
stands atop a ramp that
has a flat portion and he
is displaced (by a small
amount) to the left or
right, he won’t accelerate
away.
U=0
If he was displaced (slightly) to either side, he
wouldn’t go anywhere.
He does, however, have
energy due to his position.
Energy and Work
Potential
Energy
Curves
Neutral Equilibrium – occurs when a SMALL displacement in the particle results in
no net force and the particle remains at rest.
E
Total Energy
U
0
F(x) = -dU/dx = 0
x
If he was displaced (slightly) to either side, he
wouldn’t go anywhere.
F = -slope = zero = equilibrium!
Energy and Work
Last thing…I promise.
E
Total Energy
U
x
Potential
Energy
Curves
Consider a simple stable equilibrium
situation
(a skater skating back and forth in a “bowl” or a
spring oscillating back and forth).
There is a total amount of energy in
the system (due to initial
conditions).
The kinetic energy can be found by
applying the conservation of
(mechanical) energy:
E = U + KE
Energy and Work
Last thing…I promise.
Potential
Energy
Curves
E
Total Energy
U
KE
x
As the potential energy
increases, the kinetic energy
decreases. As the potential
thebekinetic
Theenergy
kineticdecreases
energy can
found by
energy increases.
The total
applying
the conservation
of
energy,
however,
is
always
the
(mechanical) energy:
E =same.
U + KE
Energy and Work
In you lab packet complete part 3
Potential
Energy
Curves
(Interpreting Potential Energy Curves).
Each individual student is responsible for the content of this PowerPoint.
Revisit this PowerPoint as needed to
reinforce the concepts discusses.
Each lab group is responsible for completing the lab portion of this activity and
submitting one write up per group.