potential energy curves notes and virtual lab activity – ap mechanics
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Potential Energy Curves
Notes and Virtual Lab Activity – AP Mechanics
Energy and WorkClick 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.
LabPart 2
Energy and WorkWe already know that
areadxxFW
x
x
2
1
)(
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 work we do to the ball
would be called the work applied (because our applied force is acting through some distance).
Fapp
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).
h
Potential Energy Curves
Energy and Work
h
Fapp
mg
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:Wapp= +∆PE= +∆U
Potential Energy Curves
Energy and Work
h
Fapp
mg
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.
This work (Wg= -mgh) is opposing the increase in the potential energy so we write:
Wg= -∆PE= -∆U
Potential Energy Curves
Energy and Work
h
Fapp
mg
Let us look at this in another way.What happens when we let go of the ball?
Surprise!The ball falls.
As it falls gravity does POSITIVE work on the ball and the potential energy
DECREASES.
+Wg= -∆PE= -∆U
The FIELD will ALWAYS WORK to REDUCE the POTENTIAL ENERGY!
Potential Energy Curves
Energy and WorkSo now we know…
areadxxFW
x
x
2
1
)( +Wfield= -∆PE= -∆U
Or focus is with a gravitational field, but this is true for any type of field OR restoring force.
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.
areadxxFU
x
x
2
1
)(
Potential Energy Curves
Energy and Work
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?
areadxxFU
x
x
2
1
)(
Just go the opposite way….…the reverse process of the antiderivative is the derivative.
slopedx
dUxF
)(
Potential Energy Curves
Energy and Work
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.
areadxxFU
x
x
2
1
)(
The opposite of the slope of a potential energy vs position graph gives the force acting on that particle.
slopedx
dUxF
)(
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 to identify points of equilibrium.
Energy and Work
0)(
slopedx
dUxF
Potential Energy Curves
Stable Equilibrium – think back to the pHet Skater Lab. Due to the starting position of the skater, there was a certain total amount of energy available to the system.
Energy and Work
x
Total Energy
U
E
As the skater moved, her
potential energy increased and
decreased.
0
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).
Energy and Work
x
Total Energy
U
E
0
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. slopedx
dUxF
)(
F(x) = -dU/dx = -slopeBecause the slope is negative, the force is positive.
When the skater is at x=0 the slope is
zero; this represents an equilibrium point (which happens to
be stable).
F(x) = -dU/dx = 0
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).
Energy and Work
x
Total Energy
U
E
0
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.
He does, however, have energy due to his
position
If he was displaced (off of either side) his potential energy would decrease.
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).
Energy and Work
x
Total Energy
U
E
0
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.slope
dx
dUxF
)(
F(x) = -dU/dx = -slopeBecause the slope is positive, the force is negative.
When the skater is at x=0 the slope is zero;
this represents an equilibrium point
(which happens to be unstable).
F(x) = -dU/dx = 0
Potential Energy Curves
Neutral Equilibrium – occurs when a SMALL displacement in the particle results in no net force and the particle remains at rest.
Energy and Work
xTotal Energy
E0
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.
He does, however, have energy due to his
position.
If he was displaced (slightly) to either side, he wouldn’t go anywhere.
U=0
U
Potential Energy Curves
Neutral Equilibrium – occurs when a SMALL displacement in the particle results in no net force and the particle remains at rest.
Energy and Work
xTotal Energy
E0
If he was displaced (slightly) to either side, he wouldn’t go anywhere.
F = -slope = zero = equilibrium!
U
F(x) = -dU/dx = 0
Potential Energy Curves
Energy and Work
E
x
Total Energy
U
Potential Energy Curves
Last thing…I promise.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
E
x
Total Energy
UKE
Potential Energy Curves
Last thing…I promise.
The kinetic energy can be found by applying the
conservation of (mechanical) energy:
E = U + KE
As the potential energy increases, the kinetic
energy decreases. As the potential energy
decreases the kinetic energy increases. The
total energy, however, is always the same.
Energy and WorkPotential Energy Curves
In you lab packet complete part 3(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.
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