UNIVERSITY OF GAZIANTEP
DEPARMENT OF ENGINEERING OF PHYSICS
EP 135 GENERAL PHYSICS LABORATORY I
REPORT FOR EXPERIMENT 1
MEASUREMENT OF GRAVITATIONAL ACCELERATION
WITH SIMPLE PENDULUM
(don not write this: Error propagation strategy)
Group I
Isaac Newton
Albert Einstein
Werner Heisenberg
Abdus Salam
Date of Experiment :
Date of Submission :
Deadline :
18.11.2011
22.11.2011
25.11.2011
Lab Assistant(s):
Res. Ass. Hüseyin Toktamış
Res. Ass. L. Hasan Çite
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1. OBJECTIVE
The purpose of the experiment is to determine the gravitational acceleration, g, by
measuring the period of a simple pendulum.
2. THEORY
The simple pendulum is an example mechanical system that exhibits periodic motion.
It consists of a particle-like bob of mass m suspended by a light inextensible string of length L
that is fixed at the upper end, as shown in Figure 1. The motion occurs in the vertical plane
and is driven by the gravitational force.
Figure 1. A simple pendulum
The acceleration due to gravity for small angle swings of a simple pendulum (small dense bob
on a light inextensible string) is given by[1]:
𝑔 = 4π2
𝐿
𝑇2
(1)
where T is the period of the oscillation for the pendulum.
3. EXPERIMENTAL SETUP AND EQUIPMENTS
Apparatus: String, stop watch, meter stick, bob, support rods.
The pendulum consisting of the string and
bob is attached to meter stick via support
rods as shown in Figure 2. The bob is
manually pulled such that the pendulum
makes a small angle (such as 5o) with respect
to vertical direction. The time for the
required number of oscillations of the
pendulum is recorded by a stop watch. The
period is calculated by dividing the total
measured time to number of oscillations
counted.
Figure 2: The experimental setup
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4. PROCEDURE
i. Make a simple pendulum by using the support rods, bob, and string as shown in Figure 2.
ii. Construct a data table to record the following data for each trial.
Length (L), Initial angle (θ), Number of oscillations (N), Total time taken for the number
of oscillations (t), Measured period (T=t/N).
iii. Use stop watch to measure the time it takes the pendulum to complete 10 oscillations
iv. Record your measurements and calculated values for this trial in each column of the data
table. Be sure to include uncertainty in each measurement and calculated value. For this
experiment, L and t have uncertainties (of ±0.5 mm and ±0.005 s respectively) that will
propagate in calculating error of g.
v. For the initial angle 5o, repeat the experiment for the lengths L = 0.8, 1.0, 1.2 m.
vi. For each value of L, evaluate the value of gi (i = 1 ,2, 3) and corresponding standard
deviations σi using error propagation formula. Record the results, in the form of gi ± σi.
vii. Calculate your final result using these three results of independent measurements.
Record the result such that g ± σ.
viii. Compute percentage error to compare your result (g in previous case) with the very well
known measured value of 9.80665 m/s2[2].
5. RAW DATA
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6. DATA ANAYISIS
g is calculated via Eqn (1). Error propagation formula for Eqn (1):
𝐿
𝜕𝑔
1
𝜕𝑔
𝐿
𝑔 = 4π2 2 →
= 4π2 2 and
= −8π2 3
𝑇
𝜕𝐿
𝑇
𝜕𝑇
𝑇
Substituting the above partial derivatives to the error propagation formula yields:
σ𝑔 = √(
2
2
2
2
𝜕𝑔
𝜕𝑔
1
𝐿
2
2
√
𝜎 ) + ( 𝜎𝑇 ) = (4π 2 𝜎𝐿 ) + (−8π 3 𝜎𝑇 )
𝜕𝐿 𝐿
𝜕𝑇
𝑇
𝑇
(2)
or
𝜎𝐿 2
𝜎𝑇 2
σ𝑔 = 𝑔√( ) + 4 ( )
𝐿
𝑇
(3)
where the maximum uncertainties for the length and time measurements are
𝜎𝐿 = 0.5 mm = 5 × 10−4 m and 𝜎𝑇 = 5 × 10−4 s.
For L = 0.80 m
𝑔 = 4π2
0.8
m
= 9.8021 2
2
1.795
s
0.0005 2
0.0005 2
and σ𝑔 = 9.8021√(
) + 4(
) = 0.0082 m/s2
0.80
1.795
For L = 1.00 m
𝑔 = 4π2
1.0
m
= 9.8107 2
2
2.006
s
0.0005 2
0.0005 2
and σ𝑔 = 9.8107√(
) + 4(
) = 0.0069 m/s 2
1.00
2.006
For L = 1.20 m
𝑔 = 4π2
1.2
m
= 9.7969 2
2
2.199
s
0.0005 2
0.0005 2
and σ𝑔 = 9.7969√(
) + 4(
) = 0.0060 m/s 2
1.20
2.199
All data analysis results are summarized in Table 1.
Table 1: Measured values and the error propagation results for different lengths
Pendulum length
Time for 10 oscillations
Period
Measured value
L (m)
t(s)
T = t/N (s)
g (m/s2)
0.80 ± 0.0005
17.95 ± 0.005
1.7950 ± 0.0005
9.8021 ± 0.0082
1.00 ± 0.0005
20.06 ± 0.005
2.0060 ± 0.0005
9.8107 ± 0.0069
1.20 ± 0.0005
21.99 ± 0.005
2.1990 ± 0.0005
9.7969 ± 0.0060
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7. RESULTS
For the three different results given in Table 1, the final combined result and its standard
deviation can be calculated by:
∑ 𝑔𝑖 /𝜎𝑖2 9.8021⁄0.00822 + 9.8107⁄0.00692 + 9.7969⁄0.00602
𝑔=
=
= 9.8027 m/s 2
∑ 1/𝜎𝑖2
1⁄0.00822 + 1⁄0.00692 + 1⁄0.00602
𝜎=
1
√∑ 1/𝜎𝑖2
=
1
1
1
√ 1 2+
+
2
0.0082
0.0069
0.00602
= 0.0040 m/s2
So the measured gravitational acceleration is:
9.8027 ± 0.0040 m/s2
The percentage error between the world best value and this measurement is
𝑃𝐸 =
|9.80270 − 9.80665|
× 100% = 0.04 %
9.80665
8. CONCLUSION
By means of the simple pendulum, the gravitational acceleration was measured as 9.8027(40)
m/s2. To do that, the period of the pendulum was computed from 10 oscillations for three
different lengths. The measured g was reasonably close to the world’s best accepted value
listed in [2]. The percentage error between our measurements and the accepted value was
found to be 0.04%.
9. REFERENCES
[1]. “Physics for Engineers and Scientist”, Serway, 6th Ed. Chapter 15.
[2]. http://physics.nist.gov/cuu/Constants/index.html - Fundamental Physical Constants.
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