Measurement of Gravitational Acceleration
I. Objectives
(1) Measure the gravitational acceleration g by using air track which can provide an almost
friction-free environment.
(2) Understand how to use photogate to measure the velocity and acceleration of an object.
II. Theory
A. Horizontal movement
Let’s consider an object with mass m1 which is pulled by a force T from the string and another
object with mass m2 which is pulled down by gravitational force m2g and pulled up by a force
T from the string as shown in Fig. 1. These two objects will have an acceleration a and the
system will follow the relation below if the friction is neglected:
ì T = m1 a
í
îm 2 g - T = m 2 a
We can obtain a =
…………. (1)
m2
m + m2
g or g = 1
a from the above equations. Based on this
m1 + m 2
m2
design, we can make a to be measurable by adjusting the m1 and m2. Then, we can obtain the
gravtitaional acceleration g from a.
T
Figure 1. The schematic of measuring gravitational acceleration with m1 moving horizontally.
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(2) Inclined-up movement
Let’s put the system in Fig. 1 on an inclined plane, as shown in Fig. 2. With this design, we
will not only obtain a smaller a, but also reduce the friction on m1 by lowering the normal
force from the plane. The equation of motion of the system is:
ìT－m1gsinq = m1a
í
î m2g - T = m2a
………(2)
From the above two equations, we can get the relation g =
m1 + m 2
a . Therefore, the
m 2 - m1 sin q
gravitational acceleration g can be obtained from measuring the acceleration a.
Figure 2. The schematic of measuring gravitational acceleration with m1 moving on a inclined-up plane.
III. Method
In this experiment, we use a linear air track to provide an almost frictionless environment, then
we use the track car with a “card” to pass through the photogates. We can calculate the velocity v
= Δx/Δt of the track car, where Δx is the length of the card and Δt is the time for the card passing
through two photogates. Placing two sets of photogates on both sides of air track, so we can obtain
two velocities at these two locations. Then, the acceleration can be obtained by taking the
difference and divided by the time difference. The details as follows:
(1) The track car is accelerating to a velocity by a force from the rest (both the initial position and
!
velocity are 0). The position x as a function of time can be written as 𝑥(𝑡) = " at2 .
(2) When the front end of the card on track car reaches the first photogates at x1, the card will
block the light in the photogates. Then, the photogate will send a signal to computer and record
the time as t1. When the track car keeps moving until the back end of the card leave the position
2
x1, the photogate will detect the light again. Then, it will send another signal to the computer
and record the time as t1 + Δt1. So, at this moment, the front end of the track car reaches the
position of x1 + Δx1, here Δx is the length of the card (10 cm). This experiment is using the
time difference Δt1 and the length of the card Δx to calculate the velocity of the track car 𝑣! =
∆#!
∆t1
at position x1. But how large the discrepancy between the theoretical value and the
measured value will be? Let’s estimate it:
1
1
1 ∆𝑡!
∆𝑥! = 𝑥(𝑡! + ∆𝑡! ) − 𝑥(𝑡! ) = a(𝑡! + ∆𝑡! )" − a(𝑡! )" = a𝑡! ∆𝑡! (1 +
)
2
2
2 𝑡!
𝑖. 𝑒.
𝑣! =
∆#!
∆%!
! ∆%!
= a𝑡! (1 + "
%!
) ………(3)
This means that the measured v1 will be close to the theoretical value at1 only when ∆𝑡! <<
𝑡! . In the other words, the position of the first photogate x1 must be away from the initial
position (we use 20 cm in this experiment as shown in the following procedure) which also
means that the measurement will be accurate when t1 is large enough. On the other hand, the
measured velocity will be not accurate if the velocity of the track car is too slow, Δt1 is too
large.
(3) Similarly, the velocity of the track car reaches the second photogate can be measured as
𝑖. 𝑒. 𝑣" =
∆#"
∆%"
! ∆%"
= a𝑡" (1 + "
%"
) ………(4)
Same as v1, the measured v2 will be close to the theoretical value at2 only when ∆𝑡" << 𝑡" .
Usually, this requirement will be automatically satisfied when the requirement for the first
photogate is satisfied since t2 > t1. However, since the v2 is faster and the length of the card is
fixed to 10 cm, this means the measured ∆𝑡" will be shorter and it might introduce a larger
uncertainty.
(4) Using the measured v1 and v2 from two photogates sets, and Eq. (3) and (4), the acceleration
can be obtained:
𝑎& =
'" ('!
%" (%!
! ∆%" (∆%!
= a(1 + "
%" (%!
) ………(5)
Here 𝑎& < a since ∆𝑡" < ∆𝑡! . However, we will obtain 𝑎& → a (the measured acceleration
will be close to the theoretical value) when𝑡" − 𝑡! >> ∆𝑡! − ∆𝑡" . This also means that t2 – t1
is related to the distance between two sets of photogates, so we take 40 cm in this experiment.
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IV. Apparatus
(1) A linear air track, a track car, a weight plate, weights, a block of wood, a pulley, and nylon
string.
(2) An adjustable air blower and a pipe.
(3) A computer, 2 photogates, a card, a computer’s power supply, and 10 connectors.
(4) A scale.
V. Experimental Setup
Figures 3 and 4 are the experimental setup for the gravitational acceleration experiment and the
wiring diagram of the photogate and computer, respectively.
photogate
photogate
Air track
Track car
Adjustable air
blower
Pulley, card,
Wood
Computer
Weights and
Computer’s power supply
others
Figure 3. The experimental setup for the gravitational acceleration measurement.
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Photogate 2
Signal for photogate
Photogate 1
Power for photogate
Computer’s power supply
Computer
Figure 4. The wiring diagram for the photogate and computer.
VI. Procedure
A. Using the horizontal movement to measure g:
1. Follow Figs. 3 and 4 to set up the experiment (Note: the side for pulley of the air track
needs to be exceeded about 3 cm from the desk to avoid the nylon string touching the desk.),
then weight the track car with the hook and the 10 cm card. Record the number and list
them in the lab report.
2. Place the photogate 1 and photogate 2 on the positions where are away from the pulley
about 70 cm and 110 cm, respectively, so the distance between these two photogates is
about 40 cm. (Note: you must turn on the adjustable air blower first and set it to 5. Then,
adjust the height of the photogate, so the track car can pass through it and block the light.)
3. Place the track car in-between two photogates and adjust the level button until the track
car can float on the air track statically. This means the air track is near to the perfect
condition. (Note: the track car probably won’t be able to “completely” stable, but at least
need to let it move back-and-forth in a very small region. It will be one of the most
important effects to the experimental results.)
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4. Tie the nylon string to the track car (on the hook), and place weight on the other side
(holder and weights), then place the nylon string in the pulley. (Tip: use your finger to roll
the pulley and make it spin to ensure it is not stuck.)
5. Place the track car at the position of 130 cm from the pulley (about 20 cm away from the
photogate 1), then adjust the pulley’s position to let the nylon string to be parallel to the
air track.
6. Slowly let the track car pass through the photogate1 and photogate 2, then move it to the
position of 50 cm (about 20 cm after passing through the photogate 2). Observe and adjust
to make sure the weights will not touch anything or the ground during the falling.
7. For each measurement, place the track car in the position of 130 cm from the pulley, and
then set up the computer:
a) When the computer’s screen shows <S> select mode, keep pushing <S> until it
shows “Acceleration” (this means for measuring the acceleration).
b) Push <D> and the screen will show <S> select number, then push <S> to select
readings = 1 (this means for measuring 1 reading).
c) Push <D> and the screen will show <S> mask size, keep pushing <S> until it shows
size in cm = 010.
d) Push <D> and the screen will show <G> when ready, then push <G> to select waiting.
After the track car passes through the two photogates, the screen will show display.
e) Push <D> to get the acceleration reading.
Release the track car, and use you hand to stop it at the position about 20 cm after passing
through the photogate 2 to prevent damage the equipment.
** The computer settings **
§
size in cm = 010 means the length of the object passing through the photogate is 10
cm. The computer will automatically record the time for this 10 cm object passing
through and calculate v1, v2, and the time difference Dt = t2 - t1 , then we can obtain
the acceleration a ' =
§
v 2 - v1
.
Dt
When using the computer to make multiple measurements, we can use the built-in
function to avoid the repeated setting (if we didn't turn off the power). After the last
reading is recorded, puch <D> and the screen will show <G><S><D>, then push <G>
and it will show waiting, then you can conduct the measurement again. (If you push
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<D> again, it will show the previous reading.)
8. Record your measurements in the chart and repeat it for 5 times.
9. Take the average of the 5 measurements and use Eq. (1) to calculate g and its error.
10. Try to change the initial distance to the photogate 1 and discuss the effect on the
experimental result.
11. Discuss your results.
Example chart：
The mass of track car and card (𝑚! ) = ______ g,
The mass of the holder and weight (𝑚" ) = _______ g.
Measurement
1
2
3
4
5
Avg.
STD.
"
a (m/𝑠 )
Avg. of g =________________m/𝑠 "
Error=
|*+,-.,*/012 4125, (6,175.,8 4125,|
%9:;<:%=>?@ '?@A:
× 100% =
B. Using the inclined-up movement to measure g:
1. Use the same experimental setup with a block of wood placed under the air track (on the
side of the pulley), then measure the distance L between two legs of the air track and the
height h of the wood. Then, we can immediately get sin θ =
h
. Record L, h, and sinq.
L
2. The distance between two photogates reminds the same but needs to adjust the vertical
position properly.
3. Adjust the height of pulley to ensure that the nylon string and the air track are in parallel.
4. Repeat Step 7 and 8 in Procedure A and record the acceleration in the chart.
5. Take the average of 5 measurements and use Eq. (2) to calculate g and its error.
6. Discuss your results.
Example chart:
The distance between two legs of the air track L = _______ cm,
The height of the wood (the average value of 6 positions) h = _______ cm,
sin θ =
h
= ____________.
L
The mass of track car and card (𝑚! ) = ______ g,
The mass of the holder and weight (𝑚" ) = _______ g.
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Measurements
1
2
3
4
5
Avg.
STD.
"
a (m/𝑠 )
Avg. of g =________________m/𝑠 "
Error =
|*+,-.,*/012 4125, (6,175.,8 4125,|
%9:;<:%=>?@ '?@A:
× 100% =
VII. Cautions
(1) The adjustable air blower needs to be turned on before the track car is placed on the air track
and this only can be turned off when the track car is removed from the air track.
(2) You need to be careful all the time, especially do not damage the surface of the air track.
(3) Please make sure the two buffers on the track car to have equal weight for balance.
(4) Please avoid the nylon string touch the desk when the weight is falling vertically.
VIII. Questions
(1) How to improve this experiment? Is there any way to make the measurement to be more
accurate?
IX. References
(1) University Physics: 9th edition H.D. Young (Chap 5 and Chap 12)
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Self-evaluation check list: Measurement of Gravitational Acceleration
After completing the experiment, use this check list to evaluate your grasp of the experiment. If
you have items that mark as “completely lost” or “very vague”, please re-examine the
experimental procedure, re-read the lab manual, or consult your TA or instructor to improve your
understanding of the content.
Item
Fully
Mostly
understood
understood
Vague
Completely
lost
1. Know how to use photogate to measure
the velocity of an object
2. Know how to use photogate to measure
the acceleration of an object
3. Know the design principle behind using
a photogate to measure velocity
4. Know the sameness and the difference
between the method in this experiment
and that using Tracker
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Editor: Department of Physics,
National Cheng Kung University
Revision date: 2022/09
By Yi Yang