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3/20/2017
PHY 133 Lab 5 ­ Conservation of Energy [Stony Brook Physics Laboratory Manuals]
Stony Brook Physics Laboratory Manuals
PHY 133 Lab 5 - Conservation of Energy
The purpose of this lab is to experimentally verify the conservation of mechanical energy. To do this, we will examine the conversion
of gravitational potential energy into translational kinetic energy for an isolated system of an air-track glider and a falling mass.
Equipment
air track (with picket fence)
glider
photo gate (mounted on top of the glider)
interface box (photo gate → computer)
computer
string
10g or 20g mass
Introduction
For an isolated system, the total energy must be conserved. (If no energy enters or leaves a system, then the total energy in the
system remains constant, although it may be converted from one form to another.) In this experiment, we will examine the law of
conservation of total mechanical energy in a system by observing the conversion from gravitational potential energy to translational
kinetic energy, using a glider on a frictionless air track that is pulled by a falling mass. The apparatus is called an “air track” because
an air “cushion” reduces the friction between the glider and the track surface so much that we neglect friction altogether. Hence, we
consider the glider-mass system to be isolated from friction. The position of the glider as a function of time can be accurately
recorded by means of a photogate device.
In this experiment, the glider (of mass M ) on the air track and the attached falling mass m both gain kinetic energy due to an equal
loss of potential energy experienced by the falling mass.
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PHY 133 Lab 5 ­ Conservation of Energy [Stony Brook Physics Laboratory Manuals]
The kinetic energy of the glider-mass system, when moving at velocity v, is given by
1
1
1
2
2
2
KE =
Mv +
mv =
(M + m) v . Therefore, the change in the kinetic energy of the system between two points
2
2
2
during its motion may be expressed as:
1
ΔK E = K Ef − K Ei =
2
(M + m) vf
2
1
−
2
(M + m) vi
2
1
=
2
(M + m) (vf
2
− vi
2
The potential energy of the glider-mass system, when the small mass has a height h above the floor, is given by P E =
Therefore, the change in the potential energy ΔP E of the system, when the height h of the falling mass m changes by
Δh = h f − h i , is given by:
ΔP E = P Ef − P Ei = mgh f − mgh i = mg (Δh)
)
mgh
(1)
.
(2)
Note that Δh will be negative in this experiment, since the falling mass's final height hf is less than its initial height hi . Thus,
the system's gravitational potential energy decreases as the mass falls to the floor. The principle of conservation of energy leads
us to expect that this decrease in the system's potential energy should result in an equal and opposite increase in its
kinetic energy:
ΔK E = −ΔP E
(3)
We can also apply Newton's second law to the moving system to calculate the expected acceleration of the system as a whole, and
confirm this value as well. (Since both masses M and m are attached by a taut string, they should have the same acceleration, which
we call the “acceleration of the system.”) Because the only force moving the system is the force of gravity acting on the falling mass,
the net force should equal the weight of the falling mass, i.e., Fnet = mg. However, the net force on the system should equal the
total mass of the system times the acceleration of the system, i.e., Fnet = (M + m) a. Hence, combining these relations and
solving for the acceleration of the system, we find that:
m
a =
g
(4)
M + m
Experimental Procedure
A battery-powered photogate is mounted on the glider. When activated with the small push-button on the side of the glider, the
photogate red light-emitting diode (LED) turns on whenever the picket fence over the air track blocks the photogate beam. A light
sensor at the end of the air track receives the LED signals, and the LoggerPro program in the computer measures and records the
times when the light beam of the photogate is blocked or unblocked. Make sure that the LED on the base of the glider is facing
the receiver at the end of the track. Otherwise, no time measurements can be made.
When you release the glider-mass system, the change in height Δh of the falling mass can be measured, as well as the velocity v of
the glider-mass system. Since the mass and the glider move at the same pace, the distance the mass falls will equal the distance the
glider moves along the air track. Hence, using the picket fence distances, you can indirectly measure Δh . Similarly, since the mass
and the glider move together, the velocity values v calculated in LoggerPro using the picket fence distance and the times recorded by
the photogate will apply to both the glider and the falling mass. Thus, you can compute the sum of the potential and kinetic energies
at many moments during the motion, and verify (or dismiss!) the law of conservation of mechanical energy for this system.
First, you need to prepare your setup for data collection:
1. Your lab instructor/TA has a list of the masses for all the gliders (posted to the door at the front of the lab room). Check the
number of your glider, and obtain its mass, M , from the list of glider masses. If you cannot find your glider number, you can
also measure its mass using the digital scale in the lab room. Assume an uncertainty of σ M = 1 g for this mass, and record
these values in your notebook.
2. Determine the distance d for one picket and space on the top of the air track. (This distance is analogous to the distance of a
tape and space on the ruler from the Acceleration experiment.) To do this precisely, use a meter stick to measure the distance
10d for 10 picket and space pairs, and estimate your uncertainty (σ 10d ) in this measurement. Then, divide each value by 10
to obtain d and σ d . Record all values in your notebook.
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3. Level the air track by carefully adjusting the single leveling screw at one end of the track. Rotating the screw will tilt the track
one way or the other, so adjust it until the glider remains nearly stationary on the air track. Be sure to tighten the wing nut on
the leveling screw when the track is level, to secure your adjustment. Leveling the air track is extremely important, since
the equations above assume a frictionless, perfectly level surface.
4. Tie one end of the string to the end of the glider, and pass it over the pulley at the edge of the air track. Tie the other end of
the string to a 10g or 20g mass. Record this mass m value, and assume an uncertainty of σ m = 0.2 g. The string should
be just long enough so that the mass hangs just below the pulley when the glider is pulled back to the other end of
the air track. However, it should not be too long, and the mass should hit the floor only slightly before the glider
reaches the other end of the track after the system is released.
5. Prepare the computer for data collection. To do this, double-click the Desktop icon labeled “Exp4_xv_t2.” A “Sensor
Confirmation” window should appear, and click “Connect.” The LoggerPro window should appear with a spreadsheet on the
left (having columns labeled “Time,” “Distance,” “Velocity”) and an empty velocity vs. time graph on the right.
6. Enter your value for the picket-and-space distance d . To do this, under the “Data” tab at the top of the LoggerPro window,
click “User Parameters.” On the row labeled “PhotogateDistance1,” enter your value for d (in meters, “m”). Adjust the
decimal placement number (“Places”) and the increment (“Increment”) if necessary. Then, click “OK.”
Now, you are ready to collect data:
1. Hold the glider on the air track at the far end from the pulley, with the photogate ~3 cm before the first picket. On the
LoggerPro window, click the green “Collect” button to start a trial. Once the “Waiting for data…” text appears, release the
glider, and click the red “STOP” button just before the glider reaches the other end of the air track.
2. With a “good” set of data, you should have ~13 velocity-time pairs on the spreadsheet in the LoggerPro window, and a
straight line velocity vs. time graph should appear. If you do not get a linear graph, repeat the measurement. Of the data point
values on the spreadsheet, disregard the first data point, and copy a wide selection of ~10 data points throughout the motion
into your lab notebook.
3. For each velocity value, you also need a corresponding change in height Δh . You can define this as zero for the first data
point you record, and then use the distance traveled along the air track from that first point. Each distance should be a
multiple of your d value; for example, if your first chosen point is the 2nd data point, and your second chosen point is the 5th
data point, then Δh = − (5 − 2) d. (In general, then, the distance traveled from your 2nd data point to the nth data point
would be Δh = − (n − 2) d .) Determine this change in height Δh for each of your chosen data points.
Analysis
To calculate the change in potential energy from your first data point to every other data point, use equation (2) above. To calculate
the change in kinetic energy from your first data point to every other data point, use equation (1) above. You should also calculate
the uncertainty in each quantity, noting that the uncertainty in the change in P E or K E for each data point requires adding the
uncertainty of the initial and final energies in quadrature. For example, because ΔP E = P Ef − P Ei , then using the
−
−−−−−−−−−−−−−
addition/subtraction uncertainty rule gives: σ ΔP E
values from before, and assume that σ t
= 0
2
= √ (σ P E )
f
2
+ (σ P E )
i
. For your calculations, use your σ d , σ M , and σ m
due to the photogate's high precision.
With the data you collect from a single trial, make a plot of ΔP E vs. ΔK E and of v vs. t using the Plotting Tool provided.
Find the slope of your ΔP E vs. ΔK E plot, and compare it to your theoretical expectations based on the conservation of
mechanical energy for an isolated system. If your value is not consistent with theory, what assumptions were made that might not
hold true in the non-ideal conditions of this experiment?
Use the slope of your v vs. t plot to find the acceleration of the system (and its uncertainty), and then, (once again) use this value to
calculate an estimate of the acceleration due to gravity g. Be sure to appropriately propagate ALL uncertainties as necessary to find
m
the uncertainty σ g , including the uncertainty of
! (See the Uncertainties Quiz/Homework assignment, where this was first
M +m
mentioned.) Is your estimate for g consistent with the accepted value? What may have affected your results?
phy133/lab5conservationenergy.txt · Last modified: 2016/06/21 14:53 (external edit)
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PHY 133 Lab 5 ­ Conservation of Energy [Stony Brook Physics Laboratory Manuals]
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