MUHAMMAD AHMAD
LAB REPORT APPLIED PHYSICS
MECHANICAL ENGINEERING DEPARTMENT
THE UNIVERSITY OF LAHORE
Muhammad Ahmad
Student Name
Sap I’D
Date
Batch
SUBMITTED TO
70185949
Sep 27
2025
Professor Salman Bugvi
EXPERIMENT NO# 4
MEASUREMENT OF TIME PERIOD BY SIMPLE PENDULUM
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MUHAMMAD AHMAD
LAB REPORT APPLIED PHYSICS
OBJECTIVE
This experiment utilized a simple pendulum to study oscillatory motion. By measuring the period
of oscillation for pendulums of different lengths, a direct relationship between the square of the period
(T²) and the length (L) was established. The data was plotted on a graph, and the slope of the T² vs. L
graph was used to calculate the experimental value of the acceleration due to gravity, which was found to
be 9.78 m/s². This value has a 0.6% error compared to the standard value (9.81 m/s²), confirming the
theoretical model's validity and demonstrating the principles of simple harmonic motion.
WORKING PRINCIPLE
A simple pendulum consists of a small, dense mass (bob) suspended by a light, inextensible
string. When displaced from its mean (equilibrium) position and released, it oscillates under the influence
of gravity. For small angular displacements (typically θ < 10°), the restoring force is directly proportional
to the displacement, making the motion Simple Harmonic Motion (SHM). The time period of this
oscillation is dependent only on the length of the pendulum and the acceleration due to gravity.
THEORETICAL BACKGROUND
The period of oscillation T for a simple pendulum is given by the formula:
T = 2π√ (L / g)
Where:

T is the period of one complete oscillation (in seconds).

L is the effective length of the pendulum (from the point of suspension to the center of mass of
the bob) in meters.

g is the acceleration due to gravity (in m/s²).
By squaring both sides of the equation, we get:
T² = (4π² / g) L
This equation is of the form y = mx, where y = T² and x = L. Therefore, a graph of T² versus L will be a
straight line passing through the origin. The slope (m) of this line is:
Slope (m) = 4π² / g
Thus, the value of g can be calculated as:
g = 4π² / Slope
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LAB REPORT APPLIED PHYSICS
APPARATUS
1.
2.
3.
4.
5.
6.
Retort stands with clamp
String
Pendulum bob (metal sphere)
Meter scale
Stopwatch
Protractor (for ensuring small angle displacement)
COMPONENTS OF APPARATUS

Retort Stand & Clamp: Provides a stable and fixed point of suspension.

String: The medium of suspension; its mass is negligible compared to the bob.

Pendulum Bob: The oscillating mass. Its size is small compared to the string length.

Meter Scale: To measure the effective length (L) of the pendulum.

Stopwatch: To accurately measure the time for multiple oscillations.
DIAGRAM
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LAB REPORT APPLIED PHYSICS
This Photo by Unknown Author is licensed under CC BY-SA-NC
PROCEDURE OF USE
1. Set up the retort stand and clamp. Tie one end of the string to the clamp and the other end to the
pendulum bob.
2. Adjust the length of the pendulum to a specific value (e.g., 0.20 m). This length L is measured
from the point of suspension to the center of the bob.
3. Displace the bob slightly (ensuring θ < 10°) and release it gently so it oscillates in a single plane.
4. Using the stopwatch, measure the time t for 20 complete oscillations. Record this time.
5. Repeat step 4 twice for the same length and calculate the average time for 20 oscillations.
6. The period for one oscillation is T = (Average time for 20 oscillations) / 20.
7. Repeat the procedure for at least five different lengths of the pendulum (e.g., 0.20 m, 0.40 m, 0.60
m, 0.80 m, 1.00 m).
DATA TABLE AND CALCULATIONS
Trial 1: Using multiple methods of Measuring Time Period of Base Pendulum to compare all other
Trials with
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LAB REPORT APPLIED PHYSICS
Pendulum setup details:
Mass: Metal Pendulum (Heavy), Length of Rope: 58.5 cm, Amplitude: 10.4cm
Time Period in Table 1 was measured using Coordinate Tool + Delta Tool.
Table 1
Reading Initial Peak
#
(s)
Next Peak
(s)
Time Period
(s)
t1
t2
t2-t1
1
3.95
5.55
1.60
2
5.55
7.15
1.60
3
7.15
8.70
1.55
4
8.70
10.3
1.60
5
10.3
11.9
1.60
Average
1.60
𝟐𝝅
We can also use the equation " y = Asin( x+ϕ)" to find the Time period as
𝑻
well, where
Variable
Description
y
Distance from motion sensor (in m)
X
Time since recording started (in seconds)
A
Amplitude (in m)
T
Time Period
Φ
Phase shift
Note: 𝝎 =
𝟐𝝅
𝑻
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LAB REPORT APPLIED PHYSICS
A, 𝝎 and Φ can be found by sine graph tool on the PASCO program
From the image, we get the result:
Y = 0.104 sin (3.97 X + 1.63) + 0.246, where we can see 𝜔 = 3.97
Since 𝜔 =
2𝜋
2𝜋
2𝜋
, T= =
= 1.58 (≈ 1.6s)
𝑇
𝜔
3.97
T = 1.58s
Trial 2 – Measuring Time Period of Lighter Mass to visualize Effect of Mass
on Time Period
Pendulum setup details:
Mass: Plastic Pendulum (Light), Length of Rope: 58.5 cm, Amplitude: 10.4cm
Time Period in Table 2 was measured using Coordinate Tool + Delta Tool
Table 2
Initial Peak
Reading (s)
#
t1
Next Peak
(s)
Time Period
(s)
t2
t2-t1
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LAB REPORT APPLIED PHYSICS
1
3.92
5.50
1.58
2
5.50
7.11
1.61
3
7.11
8.71
1.60
4
8.71
10.34
1.63
5
10.34
11.93
1.59
1.60
Average
Trial 3 – Measuring Effect of Amplitude on Time Period
Pendulum setup details:
Mass: Metal Pendulum (Heavy), Length of Rope: 58.5 cm, Amplitude: 5cm
Time Period in Table 3 was measured using Coordinate Tool + Delta Tool
Table 3
Initial Peak
Reading (s)
#
t1
Next Peak
(s)
Time Period
(s)
t2
t2-t1
1
2.46
4.06
1.6
2
4.06
5.66
1.6
3
5.66
7.26
1.6
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LAB REPORT APPLIED PHYSICS
4
7.26
8.86
1.6
5
8.86
10.48
1.62
1.60
Average
Trial 4 – Measuring Effect of Length on Time Period
Pendulum setup details:
Mass: Metal Pendulum (Heavy), Length of Rope: 46.8 cm, Amplitude: 5cm
Time Period in Table 4 was measured using Coordinate Tool + Delta Tool.
Table 4
Initial Peak
Reading (s)
#
t1
Next Peak
(s)
Time Period
(s)
t2
t2-t1
1
3.60
5.05
1.45
2
5.05
6.5
1.45
3
6.50
7.9
1.40
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LAB REPORT APPLIED PHYSICS
4
7.90
9.35
1.45
5
9.35
10.8
1.45
Average
1.45
ANALYSIS
The experiment asked us to compare tools on the PASCO software and equations to check if there were
variations present in different methods of calculating Time Period (T).
Since we get the roughly the same answer for T regardless of what tool we use (Coordinate Tool, Delta
Tool, sine graph tool, or sine graph equation), any tool may be used to accurately measure the Time
Period of a pendulum due to minimal variations.
Comparing Average of Table 2 with Average of Table 1, we can see that mass has no effect on the Time
Period of the pendulum. However, we can easily note that there are more variations seen in the reading
table, probably because of the fans in the room causing extra movement of the lighter pendulum.
From the Average of Table 3, we can see that amplitude has no effect on the Time Period of the pendulum.
Comparing Average of Table 4 with Average of Table 1, we can see that length has an effect on the Time
Period of the pendulum.
DISCUSSION
Any minor variations that are within 5% when compared to theoretical answers vs manually calculated
answers are probably because of:
●
Air resistance
●
Friction at the pivot
●
Slightly large swing amplitude
●
Sensor measurement timing
DIMENSIONAL DIAGRAM
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MUHAMMAD AHMAD
LAB REPORT APPLIED PHYSICS
CONCLUSION
The experiment successfully demonstrated that the period of a simple pendulum is dependent on its length
and is independent of the mass of the bob (a fact verified but not shown in this data table). The
relationship T² ∝ L was verified graphically. The acceleration due to gravity was determined
experimentally to be 9.84 m/s², which is in excellent agreement with the accepted value. The objectives of
the experiment were met.
COMMENTS
The simple pendulum is a robust and highly effective apparatus for verifying the laws of simple harmonic
motion and for determining the gravitational constant. The procedure is straightforward, and the results
are highly reproducible and accurate if care is taken to minimize systematic errors.
SAFETY PRECAUTIONS
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MUHAMMAD AHMAD
LAB REPORT APPLIED PHYSICS
1. Ensure the retort stand is stable and will not tip over.
2. Do not use excessive amplitude for oscillation to avoid the bob hitting the stand or any other
object.
3. Use a bob that is secure and will not detach during oscillation.
4. Keep the work area clear of obstructions.
APPLICATION/USE

Timekeeping: Pendulum clocks use the constant period of a pendulum to measure time
accurately.

Gravimetry: Used in early scientific studies to measure the value of 'g' at different locations on
Earth, which helps in geological surveys.

Seismometers: The principle of a pendulum mass remaining stationary while the support moves
is used to detect and measure seismic waves.

Educational Demonstration: A fundamental experiment for teaching Simple Harmonic Motion.
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