Lab 7: Conservation of Energy
Name: __________________________
Goals:
- Write down the potential and kinetic energy in a variety of mechanical systems.
- Use conservation of energy to predict the behavior of several mechanical systems.
Solving problems with energy methods:
“Energy” is really just a mathematical construct that is convenient for solving physics problems. Any problem that is solved by
using energy could also be solved with Newton’s laws, but energy methods are often much easier.
Energy is defined so that the total energy of a system never changes (at least in the absence of friction). In physics lingo, we say
that energy is a conserved quantity. To solve problems involving energy, we simply set the initial energy of the system equal to the
final energy, and then solve for a desired variable.
Kinetic and Potential Energy:
Energy comes in two varieties: kinetic and potential.
Kinetic energy is the energy of motion. Any moving object has kinetic energy, and the energy can be computed with the formula:
Kinetic energy: KE 
1 2
mv
2
Potential energy is the energy an object has by virtue of its position. For example, we say that a rock on top of a hill has some
potential energy because it can spontaneously roll down the hill, thereby converting its potential energy into kinetic energy.
We include two types of potential energy in this course: gravitational potential energy and spring potential energy.
Gravitational potential energy is the energy an object has because of its height (like the rock on top of the hill), and it can be
computed by the formula:
Gravitational potential energy: PE g  mgh
Note that the “h” in this formula depends on our choice of origin (i.e., the height we call zero). However, all that matters physically
is a change in potential energy from one state to the next, so the choice of origin doesn’t affect the result.
Spring potential energy is the energy stored in a spring that is compressed or stretched (for example, the spring guns for the
projectile motion lab were storing spring potential energy before you pulled the trigger). Spring potential energy is computed by the
formula:
1
Spring potential energy: PEs  kx 2
2
In this formula, k is called the spring constant, and it is a measure of the stiffness of the spring (we will measure a spring constant
during this lab!). x is the compression or stretch distance from the natural length of the spring.
Ideal Springs:
In this course, we assume that all spring are “ideal”, meaning that they follow Hooke’s Law: the force required to stretch the spring
is proportional to the stretch distance (in other words, it takes twice as much force to stretch the spring twice as far).
Hooke’s Law: F  kx
In this formula, k (the spring constant) is a measure of the stiffness of the spring, while x is the displacement from the natural length
of the spring.
In this lab, we will have to use a modified form of Hooke’s Law. Our spring doesn’t have a meaningful “natural length” because
the coils are all touching each other before we stretch the spring. Therefore, we will have to measure two different stretch lengths
for two different forces and then find k from the result:
F1  kx1 and F2  kx2  F1  F2  k  x1  x2   k 
F
.
x
One more note about springs: if we hang a mass vertically from a spring, then gravity sets a new equilibrium length for the spring
(the length at which the mass just sits there). As it turns out, we can (and must!) completely ignore gravity in our calculations by
using this position as the equilibrium length of the spring. In other words, we can pretend that we are looking at a horizontal
spring/mass system after the new equilibrium position has been set.
There are four stations set up around the room – each one exploring conservation of energy in a different way. At each station, you
will take data with the position sensors in order to quantitatively verify the conservation of energy. Each student should print a
graph from each station.
STATION I: Car on a ramp.
At this station, we have a cart on a tilted ramp that is pulled up the ramp by a hanging mass. The cart has a mass of about 500 g, and
the hanging mass is about 100 g.
As the hanging mass falls through a distance, d, its gravitational potential energy decreases (because its height is getting smaller).
This decrease in potential energy is matched by an increase in the potential energy of the cart (it is moving higher above the ground)
and an increase in the kinetic energy of both objects.
To solve the system, we write down the total energy (potential and kinetic) in the initial state (before the mass is released) and then
we write down the total energy in the final state (after the mass has fallen a distance of d). We can set these two quantities equal to
express energy conservation for the system – then we can use the equation to predict the velocity of the system.
1. Find the equation expressing energy conservation for the system in terms of both masses (m and M), the final velocity v, the drop
distance d, the acceleration of gravity, g, and the angle of the ramp,  . Show your work clearly.
d
M
m

d
m
Energy conservation equation: __________________________________
2. Measure the angle,  , using trigonomety.
Opposite side: _________ m
Hypotenuse:
_________ m
Angle: _________
3. Now we can use the energy conservation equation to make a prediction: how fast will the masses be moving after M has fallen
through a distance of d  50 cm ? Simply plug in all the known quantities into the equation from part 1, and solve for v:
v predicted  ________________ m/s
Now perform the experiment: let the mass fall through at least 50 cm. Print your position and velocity graphs, and figure out the
velocity of the system after the hanging mass has fallen by 50 cm. You can mark your graphs to indicate the relevant moment(s).
vmeasured  _________________
Percent difference =
vmeasured  v predicted
v predicted
100  __________________
STATION II: Bouncing ball.
At this station, we have a position sensor configured to measure the height of a ball as it experiences free fall, then hits the floor
and bounces back up.
As the ball falls to the floor, its gravitational potential energy decreases, and its kinetic energy increases to compensate for the
loss in potential energy. We can predict the speed of the ball when it hits the ground by using energy conservation – we write the
total energy for the ball before it falls, and then write the total energy when it hits the ground. We set these two expressions of
total energy equal in order to solve the problem.
1. Find the equation expressing energy conservation for the system in terms of the mass m , the initial height h , the final velocity,
v, and the acceleration of gravity, g:
m
h
Energy conservation equation: __________________________________
2. Now we can use this equation to make a prediction: how fast is the ball moving when it hits the ground? Simply plug in all the
known variables into the equation above, then solve for v:
v predicted  ________________
3. Now perform the experiment: let the ball fall to the ground as you record the data on your computer. The position sensor
should be set at the maximum sampling rate. Print your data off the computer, and figure out the speed of the ball when it hits
the ground. Your data should clearly include at least three bounces!
vmeasured  _________________
Percent difference =
vmeasured  v predicted
v predicted
100  __________________
4. When the ball hits the floor, it slows to an (instantaneous) stop, then bounces back up (but not as high as it started). This brings
up a couple interesting questions:
Where did all the kinetic energy go when the ball hit the floor?
Why doesn’t the ball bounce back up as high as it started; i.e., what accounts for the missing potential energy?
STATION III: Bowling ball pendulum.
At this station, we have a giant pendulum made from a bowling ball and a cord. As the ball swings back and forth, its gravitational
potential energy is converted into kinetic energy, then back into potential energy, and so on. The ball is moving fastest when the
potential energy is at its lowest.
We can predict the maximum speed of the ball by using energy conservation – we write the total energy for the ball before we
release it, and then write the total energy when it reaches the lowest point in its path. We set these two expressions of total
energy equal in order to solve the problem.
1. Find the equation expressing energy conservation for the system in terms of the mass m , the initial height h , the acceleration of
gravity, g, and the final velocity v.
h
v
Energy conservation equation: _____________________________
2. Now we can use this equation to make a prediction: what is the maximum speed of the ball? Simply plug in all the known
variables into the equation above, then solve for v:
v predicted  ________________
3. Now perform the experiment: let the ball drop as you record the data on your computer. Print your data off the computer, and
figure out the speed of the ball when the height is minimum.
vmeasured  _________________
Percent difference =
vmeasured  v predicted
v predicted
100  __________________
STATION IV: Spring and mass system.
At this station, we have a 200 g mass hanging from a spring. When the mass is displaced from the equilibrium position, it bounces
up and down. Oddly, we can ignore gravity in the energy analysis here – it mathematically vanishes when the equilibrium position
of the spring is set – so we only have to think about two types of energy for this problem: the kinetic energy of the mass and the
spring potential energy.
1. First, we need to find the spring constant, k, for this spring. We do this by hanging an additional 200 g mass on the first mass and
measuring the additional displacement of the spring. Compute the change in force and change in length in order to get k.
x  ___________________
F  __________________
k  __________________
2. We can predict the maximum speed of the mass by using energy conservation – we write the total energy for the mass before
we release it, and then write the total energy when it reaches the position of lowest spring potential energy (the equilibrium
position). We set these two expressions of total energy equal in order to solve the problem. Remember that gravitational
potential energy should NOT be included in your expression of energy conservation – we can pretend that gravity acts only to
set the equilibrium position of the spring/mass system.
Find the equation expressing energy conservation for the system in terms of the mass m , the initial displacement, the final
velocity, v, and the spring constant, k.
m
d
m
Equilibrium Position.
Energy conservation equation: _____________________________
3. Now we can use this equation to make a prediction: what is the maximum speed of the mass if we release it from d  10 cm ?
Simply plug in all the known variables into the equation above, then solve for v:
v predicted  ________________
Now perform the experiment: let the mass oscillate starting from 10 cm above the equilibrium position as you record the data on
your computer. Print your data off the computer, and figure out the speed of the mass when it passes through the equilibrium
position.
vmeasured  _________________
Percent difference =
vmeasured  v predicted
v predicted
100  __________________
4. Eventually, the mass will stop oscillating altogether – where does all the energy go?
5. Your predicted value should have been higher than the measured value, because we assumed the spring had no mass. Explain
(using energy) why the final velocity should be lower when you include the mass of the spring itself in the calculations.