Name: _______________________________
Conservation of Energy, p. 1/13
PRE-LAB FOR CONSERVATION OF ENERGY
Directions: Read over the lab and answer the following questions.
1. How is gravitational potential energy defined in this lab?
2. What is your Prediction 1-2? How will the kinetic energy, gravitational potential energy and
mechanical energy change as the ball falls?
3. How will the height H be used in Activity 1-3?
4. How will the slope of the force-position graph you find in Activity 2-1 be used later in the lab?
5. What is your Prediction 2-2? As the mass oscillates up and down, how will the kinetic energy,
total potential energy mechanical energy change?
adapted from RTP for P13
revised 2/2/04
Name: _______________________________
Conservation of Energy, p. 2/13
CONSERVATION OF ENERGY
Topic: Conservation of Energy, Simple Harmonic Oscillator
Objectives:
• To understand the concept of potential energy
• To introduce the concept of mechanical energy of a system
• To investigate situations where the mechanical energy is conserved and those where it is not.
• To apply the concepts of energy to the simple harmonic oscillator
Overview:
In the lab on work and energy, we defined the kinetic energy associated with the motion of a rigid,
non-rotating object: KE = 12 mv 2 where m is the mass of the object and v is its speed.
In this lab, we will explore the concept of potential energy. Suppose you lift an object at a slow,
constant speed near the surface of the earth. You must do work (apply a force over a distance) to lift
the object because you are pulling it away from the earth. The lifted object has the potential to fall
back to its original height, gaining kinetic energy as it falls. Similarly, if you stretch a spring, you must
do work. As the spring is stretched, the object attached to the end of the spring gains the potential to
return to its original position, gaining kinetic energy as it travels.
It is very useful to define the concept of potential energy as the amount of work needed to move the
object from some position (usually the initial position) to its present position. If we use this
definition, the potential energy of an object becomes smaller and smaller as it speeds up, regardless
of whether the potential energy is due to gravity or a spring (or anything else). We can now think of
kinetic and potential energy to be two different forms of mechanical energy. We define the mechanical
energy as the sum of these two energies.
Is the mechanical energy constant during the time an object falls to the earth? Is the mechanical
energy constant as a mass bobs up and down on the end of a spring? If it is, then the total amount
of mechanical energy doesn’t change and we can say that the mechanical energy is conserved.
The concept of energy conservation raises a number of questions. Does it hold quantitatively for
falling masses—Is the sum of the calculated kinetic and potential energies exactly the same number
as the object falls? Can we apply similar concepts to systems involving forces other than
gravitational forces such as systems with spring forces? What about frictional forces? In this lab, you
will begin by exploring the common expression for gravitational potential energy to see if it makes sense.
You will then measure the mechanical energy, defined as the sum of the potential energy and the kinetic
energy, to see if it is conserved when gravity is the only force acting. Next, you will explore a system
in which both gravity and a spring act on an object. You will measure the mechanical energy of this
system to see if it is conserved. Finally, you will explore the effects sliding friction or air resistance
forces have on systems. You will explore whether mechanical energy is conserved in such systems.
Writing it up:
In this handout, you will be asked to perform calculations, analyze graphs
and answer questions. It is strongly recommended that you do all the calculations and answer all the
questions as you go through the experiment. This record is for your purposes only. You will not be graded
on it. You will be graded on how well you understand the material covered in this lab, so you should
do the entire activity, discuss the questions with your partners and take careful notes.
Equipment Note:
Do not allow the heavy masses to fall on the motion detector!
adapted from RTP for P13
revised 2/2/04
Name: _______________________________
Conservation of Energy, p. 3/13
Investigation 1: Gravitational Potential Energy
Suppose that an object of mass m is lifted through a distance y. To cause the object to move upward
at constant velocity, you will need to apply a constant force upward, just equal to the gravitational
force, which is downward.
Q1-1: Based on your knowledge from lecture or from a previous lab, how much work will
you do to lift the object through a distance y. Explain.
We choose to define the gravitational potential energy, GPE, of an object of mass m to be equal to
the work done to lift it:
GPE = mgy .
Now you can use this equation to calculate the potential energy of the ball.
Activity 1-1: Measuring Gravitational Potential Energy
1. Set up the motion detector on the floor with a screen above it to protect the motion detector.
2. Configure the software to measure position and velocity at the fastest possible sampling rate.
3. Measure the mass of the ball you will drop: ____________ (Don’t forget units).
4. Create a new data column called GPE (for gravitational potential energy) calculated from
GPE = mgy . Set up to display axes like the ones below for position-time and GPE-time graphs.
5. Hold the ball from the sides or above with your hand. Starting with the ball about 0.2 m above
the detector, begin graphing position and gravitational potential energy as you raise the ball to
about 2.0 m above the motion detector.
Position (m)
6. When you get a good set of graphs, sketch the results or print out the graphs and label them.
GPE (J)
0
0
time (s)
adapted from RTP for P13
revised 2/2/04
Name: _______________________________
Conservation of Energy, p. 4/13
Q1-2: Do the two graphs look similar? Is this surprising? Explain.
Q1-3: Gravitational potential energy is always measured with respect to a particular height
where its value is defined to be zero. In this case, what has been chosen as the
reference level? In other words, at what location would its potential energy be zero?
Q1-4: Suppose that the ball is dropped and you know the velocity at a certain time. What
equation would you use to calculate the kinetic energy of the ball?
Q1-5: Suppose the ball is dropped from some height. What equation would you use to
calculate the mechanical energy?
Prediction 1-1:
As the ball falls how will the kinetic energy change? How will the
gravitational potential energy change? How will the mechanical energy change?
Activity 1-2:
Mechanical Energy
We can check the last predictions by measuring the two types of mechanical energy and their sum as
the ball falls.
1. Create new column for Kinetic Energy, calculated from the ball’s mass and speed. Create a
column for Mechanical Energy calculated from the sum of the Gravitational Potential Energy
and Kinetic Energy.
2. Set up axes to GPE and KE versus position on the same set of axes, as shown below. Plot the
Mechanical Energy on the second set of axes shown below. The position axes of the two graphs
should be aligned, as shown below.
adapted from RTP for P13
revised 2/2/04
Conservation of Energy, p. 5/13
0
GPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
0
position (m)
3. You are now ready to examine how gravitational potential energy, kinetic energy and mechanical
energy vary as the ball drops. Hold the ball about 2.0 m above the motion detector. Begin
graphing, and as soon as you hear the clicks of the motion detector, release the ball.
4. Once you get a good set of graphs, sketch the results (or print them out and label them).
5. On your graph, label the time interval during which the ball was falling.
Q1-6: How did the variation of kinetic energy and gravitational potential energy compare to
your predictions?
Q1-7: What seems to be true about the mechanical energy for this system? Does this agree
with your prediction?
Activity 1-2:
Mechanical Energy
Another system where the gravitational force is essentially the only net force is a cart moving on an
inclined ramp with little friction. You can easily investigate the mechanical for this system as the cart
moves down the ramp.
1. Set up the ramp and the motion detector as shown below. The ramp should be inclined at an
angle of about 15o above the horizontal. The friction pad should not be in contact with the
ramp.
adapted from RTP for P13
revised 2/2/04
0
GPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
0
Conservation of Energy, p. 6/13
Q1-8: The gravitational potential energy of the cart which has traveled a distance x up the
ramp is mgy, where y is the height of the cart above the table top. You should be able
to find an equation for GPE in terms of the position x measured by the motion
detector, the length L of the ramp and the elevation H of the end of the ramp. Write
that equation below. What is the reference height for this expression for potential
energy?
2. Measure the values of H, L, and the mass of the cart. Write the equation for GPE in terms of a
single constant times x, using the values of L, H, m and the acceleration due to gravity.
time (s)
L: __________
H: __________
Mass: ___________ (Units…)
GPE =
(Choose your units so that your expression gives the GPE in Joules)
3. Modify your expressions in the software’s formulas to match the correct formulas for GPE and
KE for the cart on the incline.
4. Set up axes to GPE and KE versus time on the same set of axes, as shown below. Plot the
Mechanical Energy on the second set of axes shown below. The time axes of the two graphs
should be aligned, a shown below.
Prediction 1-2:
As the cart rolls down the ramp, how will kinetic energy change? How will
the gravitational potential change? How will the mechanical energy change?
adapted from RTP for P13
revised 2/2/04
Conservation of Energy, p. 7/13
5. Hold the cart at the top of the ramp and begin graphing. When you hear the clicks of the motion
detector, release the cart, and stop it when the cart is about 0.2 m from the detector.
0
6. Sketch you graphs or print them out and label them.
Q1-9: Compare your graphs to those for the falling ball in Activity 1-2. How are they
similar and how are they different?
GPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
Q1-10: What kind of variation is there in the mechanical energy as the cart rolls down the
ramp? Does this agree with your prediction? Explain.
0
time (s)
Prediction 1-4:
Suppose the cart is given a push up the ramp and released. It moves up,
reverses direction and comes back down again. How will the kinetic energy change? How will
potential energy change? How will the mechanical energy change? Describe in words and sketch
your predictions with labeled dashed lines on the axes below.
7. Test your prediction. Hold the cart about 0.2 m from the motion detector and begin graphing.
When you hear the clicks of the motion detector, give the cart a push up the ramp. Stop when
the cart returns to its original position.
8. When you get a good set of graphs, sketch your results on the axes that follow, or print your
graphs and label them.
If you have additional time, do the Extension below to examine the effect of friction.
adapted from RTP for P13
revised 2/2/04
Conservation of Energy, p. 8/13
Extension 1-4: Mechanical Energy and Friction
0
GPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
0
Prediction E1-5:
Suppose there is also a frictional force acting on the cart. Then as the cart
rolls down the ramp, how will the kinetic energy change? How will the gravitational potential energy
change? How will the mechanical energy change? Compare your predictions to the case where
friction was very small.
1. Test your predictions. Adjust the friction pad so that there is a significant amount of friction
between the cart and the ramp, but so that the cart rolls down the ramp when released. Using
exactly the
timesame
(s) setup as before, graph KE, GPE and mechanical energy as the cart rolls down
the ramp.
2. Sketch your results on the axes below, or print out the graphs and label them.
Q1-11: Is the mechanical energy constant for the motion of the cart down the ramp with
friction?
Q1-12: If you found that mechanical energy was not conserved as the cart rolled down the
ramp, explain what happened to the missing energy.
adapted from RTP for P13
revised 2/2/04
Name: _______________________________
Conservation of Energy, p. 9/13
Investigation 2: Elastic Potential Energy
As mentioned in the Overview, it is useful to define other kinds of potential energy besides
gravitation potential energy. In this investigation, you will look at another common type, elastic
potential energy, which is associated with the elastic force exerted by a spring that obeys Hooke’s law.
You may have read in your text or heard in lecture that the magnitude of the force applied by most
springs is linear related to the amount the spring is stretched beyond its unstretched length. This is
usually written F = kx where k is called the spring constant.
You have seen in the Work and Energy lab that the work done by a force can be calculated from the
area under the force vs. position graph. Shown below is the force vs. position graph for a spring that
obeys Hooke’s law. Note that k is the slope of this graph.
Q2-1: How much work is done in stretching this spring (i.e. a spring with spring constant
k that obeys Hooke’s law) from its unstretched length by a distance x? (Hint: Look
at the triangle on the graph above).
If we define the elastic potential energy as the work done in stretching a spring, the definition will be
analogous to the way we defined GPE. In the case of a spring that obeys Hooke’s law, the elastic
potential energy (EPE) would be
EPE = 12 kx 2
In this investigation, you will see how a real spring deviates from the idealization of Hooke’s law.
You will determine the relationship between force and amount of stretch for your spring and use the
graph to determine the EPE for a spring that does not obey Hooke’s law.
adapted from RTP for P13
revised 2/2/04
Force
Name: _______________________________
Activity 2-1:
Conservation of Energy, p. 10/13
Spring Constant (almost)
In order to calculate the potential energy in a spring, you first need to determine the relationship
between force and stretch for the spring. This can be done by measuring a series of forces and
corresponding
positionstretches.
1. Set up to test your predictions using the apparatus show at the right.
You will need a force sensor, a position detector, spring, hanging mass,
and a support. Set the software to the fastest sampling rates possible.
Set the force probe for pull positive.
Prediction 2-1:
On the axes that follow, sketch a prediction for the
force vs. position graph for a spring that obeys Hooke’s law. Notice that
stretch and position are not the same. Your graph should not look like the
one on the previous page.
2. Check the calibration of the probe. Use the software to display axes for a force-time graph.
Hang the spring from the force probe and zero the probe. Then hang a known mass (preferably
a large one) and make sure the force probe reads the correct value.
3. Use the software to display axes for a force-position graph. Place a mass on the end of the
spring. Choose as large a mass as the spring can handle. Begin graphing. Gently lift the mass at
a slow, constant velocity. Make sure the motion detector “sees” the bottom of the mass (and not
your hand) as you lift the mass. Continue lifting (and graphing) until you reach the spring’s
unstretched length. Sketch the graph on the axes above or print the graph.
Q2-2: Was the force exerted by the spring proportional to the displacement of the spring?
Is the relationship proportional for any portion of the graph you studied? Explain.
4. Use the software to find the equation of the linear portion of the data. Record the fit equation
below. Record the value (and units) for the slope. The slope of the graph is closely related to the
spring constant k described in most textbooks.
Fit Equation:
____________________________ (linear portion only)
Slope (call this k ! for now):
__________________ (units…)
Q2-3: Does Hooke’s law properly describe the relationship between force and stretch for
the spring you used? Explain. How is the relationship for your spring similar to
Hooke’s law? How is it different?
adapted from RTP for P13
revised 2/2/04
Note: From Prediction 22, you might expect that
Conservation of Energy, p. 11/13
the mechanicalName:
energy_______________________________
of
the system might be
difficult to calculate, since
Q2-4: Does the relationship EPE = 12 kx 2 hold for this spring? How can the potential
the total potential energy
includes contributions
energy of this spring be calculated? Explain.
from gravitational potential
energy and elastic potential
energy. ThingsPrediction
are
2-2:
Hang a large amount of mass from the spring. The spring should stretch out
especially complicated
considerably, but not so much that the spring might fail. Start it oscillating by pulling it down a small
because the relationship
amount and letting it go. As the mass oscillates up and down, how will the kinetic energy change?
does not
How
will the total amount of potential energy (i.e. the sum of elastic potential energy and gravitation
apply to the spring
you energy) change? How will the mechanical energy change?
potential
used. However, it can be
shown mathematically that
the motion of the mass
relative to the equilibrium
position
is only
Activity 2-2: Mechanical Energy
influenced by the spring
force and not In
by this
the activity, you will explore the mechanical energy of the hanging mass on a spring as it
oscillates
around equilibrium.
gravitational force.
It can
also be shown that the
motion relative to the
equilibrium position is only
affected by the slope of the
spring’s force-position
graph near equilibrium and
not by other features of the
force-position graph. The
final result is that the total
potential energy can be
calculated using
where is the slope of the
force-position graph at
.
1. Suspend the mass from the spring and be sure it is at rest.
2. Set up axes to display position vs. time. Begin graphing and determine the equilibrium position
from the motion detector. Record the mass and the equilibrium position below.
Mass: _________________
Equilibrium position: ______________________ (Don’t forget units…)
3. Create a new data column called the total potential energy calculated using
TPE = 12 k "( x − xeq ) 2 . In Investigation 1, you used the software’s calculation tools to calculate
KE and mechanical energy. If these calculations are still available, you can modify the old
expressions. Otherwise, you will have to create new data columns for KE and mechanical
energy. Either way, the expression for the KE should reflect the correct mass for this
adapted from RTP for P13
revised 2/2/04
Conservation of Energy, p. 12/13
investigation and the mechanical energy for this investigation should be the sum of TPE and
KE.
0
4. Set up to display axes like the ones that follow for TPE-time and mechanical energy-time graphs.
TPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
0
time (s)
5. Start the mass oscillating with an amplitude of about 5 cm, and then begin graphing. Sketch your
results on the axes above or print them out and label them.
Q2-5: How did the observations of kinetic energy and total potential energy compare to
your predictions?
Q2-6: What seems to be true about the mechanical energy defined as the sum of the total
potential energy and the kinetic energy? Does this agree with your prediction?
Activity 2-3:
Mechanical Energy with Air Resistance
Prediction 2-2:
Suppose that there is also a significant amount of
air resistance in addition to the spring force acting on the mass as it
oscillates up and down. How will the total potential energy change? How
will the mechanical energy change? Compare your predictions to the
case you just examined where the air resistance was very small.
adapted from RTP for P13
revised 2/2/04
0
Conservation of Energy, p. 13/13
To test your predictions, you add a large cardboard sail to the mass.
1. Measure and record the mass of the sail.
Mass of Sail: ____________ (units…)
TPE and KE (J)
Mechanical Energy (J)
Name: _______________________________
Total of hanging mass and Sail:
__________________ (units…)
2. Attach the cardboard securely to the mass, as shown in the diagram.
3. Determine the new equilibrium position.
Equilibrium position: _____________ (units…)
0
4. Enter these new values into the potential energy and kinetic energy formulas.
time (s)
5. Start the mass and sail oscillating, and begin graphing. Sketch your results on the axes that follow
or print out the graphs and label them.
Q2-7: Is the mechanical energy constant for the motion of the mass with air resistance? Is
mechanical energy conserved?
Q2-8: If you found mechanical energy was not conserved, where did the energy go?
adapted from RTP for P13
revised 2/2/04