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.