The document describes an experiment to determine the acceleration due to gravity (g) using a simple pendulum. Measurements were taken of the time period (T) of oscillations for different pendulum lengths (L). Using the linear least squares regression method, the slope of the T2 vs L graph was found to be 0.03886, corresponding to a value of g = 10.15 m/s2. The experiment was repeated in Excel, finding a slope of 0.0328 and value of g = 9.44 m/s2, with a 3.673% difference between the methods. The regression line analysis allows calculation of g from the period-length relationship of a simple pendulum.

1. Physics Lab Page 1 of 9
AMERICAN INTERNATIONAL UNIVERSITY–BANGLADESH (AIUB)
FACULTY OF SCIENCE & TECHNOLOGY
DEPARTMENT OF PHYSICS
PHYSICS LAB 1
Summer 2022-2023 Section: C-5, Group: 08
LAB REPORT ON
Determining the acceleration due to gravity applying linear least square regression method by
using a simple pendulum.
Supervised By
DR. KAMRUN NAHAR MUKTA
Submitted By
Name ID Contribution
Date of Submission: July 11, 2023

2. Physics Lab Page 2 of 9
TABLE OF CONTENTS
TOPICS Page
no.
I. Title Page………………………………………………… 1
II. Table of Contents…………………………………………. 2
1. Theory……………………………………………………. 3
2. Apparatus………………………………………………… 5
3. Procedure…………………………………………………. 5
4. Experimental Data……………………………………….. 6
5. Analysis and Calculation………………………………… 7
6. Result……………………………………………………… 8
7. Discussion…………………………………………………. 8
8. References………………………………………………… 9

3. Physics Lab Page 3 of 9
1. Theory:
Theoretically, the period of a pendulum is independent of its mass and depends on length
according to the power of relationship.
A simple pendulum is an arrangement consisting of a metal bob connected by a light thread
and hanging vertically from a fixed support. The time required for one complete oscillation of
a basic pendulum is known as the time required for one complete oscillation is known as the
time period & it is denoted by T. L stands for effective length, which is the distance from the
point of suspension to the center of the metal bob. A basic pendulum's time period for small-
angle oscillation is,
𝑇 =2π
√𝐿
𝑔
Here, g is the acceleration due to gravity.
Rigid support
Figure 1.1: A swinging simple pendulum with an effective length L and amplitude θ.

4. Physics Lab Page 4 of 9
By squaring both sides, the time period equation of a simple pendulum can be written as,
The equation of the straight line passing through the origin is y = mx. Comparing this equation with
the equation of time period, the value of g i,e. acceleration due to gravity can be determined as here m
is the slope of the T 2
vs L graph.
If there are two types(one is dependent and another one independent) of variables x and f(x), the
linear least square regression method can be used for N number of data points to find the best fitted
line which is the regression line as the figure shows below,
Figure 1.2: Way to get the best fitted line by finding the minimum value of
We can determine the slope of the regression line using the formula
m=
∑ 𝑖𝑥𝑖𝑦 𝑖−
(∑ 𝑖𝑥 𝑖)(∑ 𝑖𝑦𝑖)
𝑁
∑ 𝑖𝑥𝑖2−
(∑ 𝑖𝑥𝑖)2
𝑁
and intercept c =y
̅ -mx
̅ , where x
̅ and y
̅ are mean value of x and y.

5. Physics Lab Page 5 of 9
In the equation of slope:
2. Apparatus:
To contrast a Simple Pendulum
1. Metal bob
2. Apiece of string
3. Stand and Clamp
Measurement of L and T:
1. Stop Watch
2. Meter Scale
3. Procedure:
1. First, we tied a light length of rope to the metal bob's hook. We measured the length of the
pendulum from the point of suspension to the bob's midpoint using a meter scale.
2. The pendulum was then given a small angle swing of less than 10 degrees. We discovered the
time period (T). To do this we calculated the total time for 20 oscillations and divided it by
20. We repeated the operation for various lengths and logged the results in table 1.1.
3. Using the Linear Least Squares Regression Method (LLSRM), we discovered the
regression line and from the value of slope we found g from the relation:
slope = 4π
2
/𝑔
4. We then displayed the same graph in Excel and calculated the value of g using the graph's
equation.

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4. Experimental Data:
Table 1.1: Time periods (T) for different lengths (L) of the simple pendulum
No.
of
Obs.
Effective
Length
L
(cm)
Time for 20
Oscillations
t
(s)
Time Period
T = t/20
(s)
𝑇²
(s²)
𝐿²
(cm²)
L.𝑇²
(cm.𝑠²)
1 110 42.22 2.11 4.4521 21000 975.4106
2 100 40.50 2.025 4.1006 10000 962.7473
3 90 38.81 1.9405 3.7655 8100 943.5818
4 80 36.78 1.839 3.382 6400 933.8478
5 70 33.97 1.698 2.8832 4900 958.4799
6 60 32.62 1.631 2.6601 3600 890.457
7 50 28.84 1.442 2.0793 2500 949.3199
Σ 560 23.2338 47600 6613.844
5. Analysis and Calculation:
Table 1.2: Finding the slope, m and intercept, c by using the linear least square regression method
N
∑ 𝑥𝑖
𝑖
∑ 𝑦𝑖
𝑖
∑ 𝑥𝑖𝑦𝑖
𝑖
2
(∑ 𝑥𝑖)
𝑖
∑ 𝑥𝑖²
𝑖
m c
7 560 23.3228 1974.641 313600 47600 0.03886 0.223
Equation: y = 0.03886 x +0.223

7. Physics Lab Page 7 of 9
A. The value of g using the LLSRM:
=
1974.641−
(560)(23.3228)
7
51600−
313600
7
= 0.03886
X
̅ =
𝑖 𝑖
∑ 𝑥
𝑁
=
560
7
=80
y
̅ =
𝑖∑ 𝑦𝑖
𝑁
=
23.3228
7
= 3.3318
Now intercept c = y
̅ -mx
̅ = 3.3318 – (0.03886) (80) = 0.223
Acceleration due to gravity by using LLSRM,
gL=
4π2
m
=
4(3.1416)2
0.03886
= 1015.91 cms-2
= 10.15 ms-2
m=
∑ 𝑖𝑥𝑖𝑦 𝑖−
(∑ 𝑖𝑥 𝑖)(∑ 𝑖𝑦 𝑖)
𝑁
∑ 𝑖𝑥𝑖2−
(∑ 𝑖𝑥𝑖)2
𝑁

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6613.844
7
=
B. The value of g from the graph of Excel:
Slope of the regression line, m = 0.0328 [comparing equation y = 0.03886x + 0.223 with y = mx +
c]
Acceleration due to gravity from the graph of Excel:
Gm
= 9.44 ms-2
C. Percentage of difference in g between Excel and LLSRM:
𝑔 ∼ 𝑔
𝐸 𝐿
𝑔
𝐸
× 100 = 9.8 – 9.44
9.8
× 100 = 3.673%
6. Result:
Method 2
Value of g (𝑚/𝑠 ) Comment
LLSRM 10.15 We have found small
difference in the value of g.
Mean 9.44
7. Discussion:
m=
∑ 𝑖𝑥𝑖𝑦 𝑖−
(∑ 𝑖𝑥 𝑖)(∑𝑖𝑦𝑖)
𝑁
∑ 𝑖𝑥𝑖2−
(∑ 𝑖𝑥𝑖)2
𝑁
1. The acceleration caused by gravity, symbolized by the letter "g," is the uniform acceleration
that the earth's gravitational pull causes in a body that is falling freely.
2. With the assistance of a length of rope, a stand, and a metal bob, a straightforward pendulum
can be simply built. The string ought to be massless and untraceable. Other than gravity, there

9. Lab Report page 9 of 9
shouldn't be any other forces produced by outside forces. To represent bob as a point, bob's
length to dimension ratio needs to be high.
3. The regression line can be obtained using the linear least square regression method (LLSRM)
by determining the minimum value of
4. For determining the slope of the regression line, the formula is and intercept 𝑐 = 𝑦 − 𝑚𝑥, where 𝑥
𝑎𝑛𝑑 𝑦 are mean value of x and y. Using these formulas we can find the slop and intercept for any
number of data.
5. This is how we can calculate the acceleration due to gravity where m = slope of the T²VS L Graph.
8. References:
 Fundamentals of Physics: Acceleration due to gravity (Chapter 13, page 360), Simple
pendulum (Chapter 15, page 425-426)
 Video Links,
 Simple pendulum: https://youtu.be/z6ZdhP167RY
 https://youtu.be/02w9lSii_Hs
 LLSRM: 1. https://youtu.be/NN62YDcr3eU
 https://youtu.be/1Q15fgz-lUk

10. Lab Report page 9 of 9