PHYSICS I LAB REPORT
COURSE CODE: PHY102L
Spring 2020-21
Experiment No.
5
Experiment Title
To determine the effect of changing the length of a
pendulum on its Period
Group No.
_
Section No.
_
NO. STUDENT NAME
STUDENT ID
1.
2.
3.
4.
Submission date
Wednesday, April 28, 2021
Objective
The basic purpose of this lab is to understand the basic concepts related to simple pendulum,
understand the type of motion occurring in it, i.e. oscillatory motion. The theoretical aspects of
simple pendulum, i.e. its time period, frequency and visually depicting the motion of simple
pendulum on position-time graph is also discussed here. This experiment also enables students to
determine the effect of changing length of simple pendulum on its time period, and thus using
mathematical correlations and graphical analysis based on these correlations, a physical constant,
i.e. gravitational acceleration is calculated.
Introduction And Background theory

Simple Pendulum:
A pendulum is an object that is attached to a pivot point, so it can swing freely. This object is
subject to a restoring force that will accelerate it toward an equilibrium position. When the
pendulum is displaced from its place of rest, the restoring force will cause the pendulum to oscillate
about the equilibrium position. In other words, a weight attached to a string swings back and forth.
A basic example is the simple gravity pendulum or bob pendulum. This is a weight (or bob) on
the end of a massless string, which, when given an initial push, will swing back and forth under
the influence of gravity over its central (lowest) point.
The regular motion of pendulums can be used for time keeping, and pendulums are used to
regulate pendulum clocks.
Figure 1 A Simple Pendulum

Assumptions while analyzing the motion of Simple Pendulum:
A simple pendulum is an ideality involving these two assumptions:
 The rod/string/cable on which the bob is swinging is massless and always remains
inextensible.
 Motion occurs in a plane.
Velocity of simple pendulum is
Vmean = Maximum,
Vextreme = 0
Acceleration of simple pendulum is
amean = 0,
aextreme = Maximum
K.E of simple pendulum is
K.Emean = Maximum,
K.Eextreme = 0
P.E of simple pendulum is
P.Emean = 0,
P.Eextreme = Maximum

Theory of Simple Pendulum:
Figure 2 Forces Acting on A Simple Pendulum
Under the above assumptions, the equation of motion of simple pendulum can be written as:
2nd Law of motion for linear motion;
𝐅 = 𝐦𝐚 − − − −𝐄𝐪(𝟏)
From figure 2, the restoring force is given by;
F = −mg sin θ − − − −Eq(2a)
As both the forces must be equal, so;
𝑚𝑎 = −𝑚𝑔 𝑠𝑖𝑛 𝜃 − − − −𝐸𝑞(2𝑏)
and for smaller angles;
sin θ ≈ θ
Where, angle is given by figure 2;
θ=
x
− − − −Eq(3)
L
where, x = maximum amplitude of the pendulum;
So; from equations 2a, 2b and 3;
a = −g𝜃
and;
a = −g
𝑥
𝐿
a ∝ −𝑥
As, g and L are constant, so acceleration must be directly proportional to the amplitude of the
body, and thus motion is said to be simple harmonic.

Time Period and frequency of Simple Pendulum:
Time period of an oscillating body is the time required to complete its one vibration. Thus, in case
of simple pendulum, it is the time required to move from an extreme to another extreme, and then
back to the same extreme position (i.e. back and forth). Frequency of the pendulum is the number
of complete swings (vibrations) per unit of time. The time period is the reciprocal of the frequency.
Mathematically, in the absence of frictional forces;
L
T = 2π√ − − − −Eq(4)
g
𝑇 2 = 4𝜋 2
𝑙
𝑔
And, from the above relations, gravitational acceleration is given by;
𝑔=
4𝜋 2 𝐿
− − − −Eq(5)
𝑇2
Finally, frequency of the pendulum is given as;
𝑓=
1 𝑔
√ − − − −𝐸𝑞(6)
2𝜋 𝐿
Apparatus
The apparatus consists of following:

PASCO Interface

Motion Sensor

Photogate Pendulum Set

Pendulum Clamp

Rod

String, spool

Measuring tape

Protractor

Large Rod Base
Figure 3 Experimental Setup; Part A
Figure 4 Experimental Setup; Part B
Observations; Calculations and Data Analysis
Part A; Effect of Length of Simple Pendulum on Time Period
Data
Table 1 Experimental Data for Part A
Effect of length
Length
Time of 10 oscillations
L (m)
0.4
0.5
0.6
0.7
0.8
t1 (s)
12.67
14.25
15.44
16.61
17.93
t2 (s)
12.61
14.1
15.41
16.67
17.87
tavg (s)
12.64
14.175
15.425
16.64
17.9
Time
Period
Time Period
Squared
T (s)
T2 (s2)
1.264
1.597696
1.4175
2.00930625
1.5425
2.37930625
1.664
2.768896
1.79
3.2041
Draw a graph of Length (L) versus (T2) by using excel sheet and calculate acceleration due
to gravity (gexp).
Graph Drawn (A plot of Length of the Pendulum (m) vs. Square of Time Period (s2))
Length (m) vs. Square of Time Period(s2)
0,9
y = 0,2509x
R² = 0,9993
0,8
0,7
L (m)
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,5
1
1,5
2
2,5
3
T2 (s2)
Figure 5 A plot of Length of the Pendulum (m) vs. Square of Time Period (s 2)
3,5
Graphical Analysis and Calculations
As we know that time period of a simple pendulum is given by;
L
T = 2π√ − − − −Eq(6)
g
Where, L = Length of simple pendulum (m) ,
T = Time Period of oscillations (s)
g = Gravitational Acceleration (m/s2)
T 2 = 4π2
𝐿
g
And, experimental gravitational acceleration is given by;
g exp = 4π2
L
− − − −Eq(7)
T2
And;
∆L
= 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐺𝑟𝑎𝑝ℎ
∆(T 2 )
Thus, experimental gravitational acceleration is given by;
g exp = 4π2 × Slope − − − −Eq(8)
As; from the graph drawn; best equation of fit comes out to be (with intercept 0 (as a direct
relationship between dependent and independent variables));
y = 0.2509x
R² = 1
Slope of the Graph = 𝟎. 𝟐𝟓𝟎𝟗 𝐦/𝐬𝟐
Thus, experimental gravitational acceleration is given by;
g 𝑒𝑥𝑝 = 4π2 × 𝑆𝑙𝑜𝑝𝑒 = 4π2 × 𝟎. 𝟐𝟓𝟎𝟗 𝐦/𝐬𝟐
g 𝑒𝑥𝑝 = 9.905 m/s2
Also;
Actual (Theoretical) Value of g is given by;
g 𝑡ℎ = 9.8 m/s2
Percentage error is given by;
%Error =
|g exp − g th |
× 100 − − − −Eq(9)
g th
So, putting values;
% Error =
| 9.905 m/s 2 − 9.8 m/s2 |
× 100
9.8 m/s 2
% Error = 1.073 %
Part B; Time Period Using Position-Time Graph of Pendulum
Data
Table 2 Experimental Data for Part B
Length L (m)
Time of one
oscillation T (s)
0.70
1.65
Calculations
As we know that time period of a simple pendulum is given by;
T = 2π√
L
g
T 2 = 4π2
𝐿
g
And, experimental gravitational acceleration is given by;
g exp = 4π2

L
T2
Now, time period is found from experimental position-time graph (the time taken by the
pendulum to cover a complete oscillation covering distance equal to wavelength (distance
between two consecutive troughs in our case).
Thus, for two troughs shown with data in figure 2,
T = 8s − 6.350s
T = 1.650 s
Thus, putting the values;
g exp = 4π2
0.70 m
(1.65 s)2
g exp = 10.151 m/s2
Also;
Actual (Theoretical) Value of g is given by;
g 𝑡ℎ = 9.8 m/s2
Percentage error is given by Eq(4) listed above;
%Error =
|g exp − g th |
× 100
g th
So, putting values;
% Error =
| 10.151 m/s2 − 9.8 m/s2 |
× 100
9.8 m/s 2
% Error = 3.577 %
Questions
1. What happens to the period of a pendulum if its amplitude ( the angle ) changes slightly,
what happens to the period if its length changes, what happens to the period if it’s mass
changes?
Ans;

As the amplitude changes slightly, the period of a simple pendulum remains same.

As the length changes slightly, the period of a simple pendulum also changes. The period
increases with an increase in the length of pendulum.

As the mass changes slightly, the period of a simple pendulum remains same.
2. Why is it important to hold the mass and amplitude constant when testing the effect of
length on the period of the pendulum?
Ans; It is important to hold the mass and amplitude constant when testing the effect of length on
the period of the pendulum because in order to determine the effect of one variable, other variables
must be kept constant.
3. Did you prove the relation between time-period and length of the pendulum? Explain.
Ans; Yes, the graph drawn between length and square of time-period is a straight-line graph
showing linear relationship between length and square of time-period of the pendulum.
𝑳 ∝ 𝑻𝟐
Or;
𝑇 ∝ √𝑙
4. What were your predictions before starting the experiment? Does your result match your
predictions before the experiment?
Ans; It was predicted that the period will increase with an increase in the length of the pendulum
and the experimental results also support this prediction. Also, the predicted value of g was 9.8
m/s2 , and in both parts of experiment, the value of ‘g’ comes out to be in close proximity with
accepted value, < 5% deviation in both cases.
5. What should you do to the length of the string of a simple pendulum to double its period?
Ans;
For time-period = T, length = L.
𝑻𝟐 𝒈
𝑳=
𝟒𝛑𝟐
For time-period = T, length = x.
(2T)2 = 4π2
x
g
𝑇 2𝑔
𝑥 = 4 ( 2)
4π
𝒙 = 𝟒𝑳
Therefore, length should be four times the original length.
6. A little girl is sitting on a swing and swings back and forth with a period of 3.0 seconds. If
she stands up on the swing the period of her swing will be.
a) Longer b) shorter c) unaffected.
Ans; The period will become shorter because as the girl stands up, the center of mass will also
shift upwards, reducing the effective length of pendulum and since:
𝑻 ∝ √𝑳𝒆𝒇𝒇𝒆𝒄𝒕𝒊𝒗𝒆
The period will decrease.
7. The acceleration of gravity gets smaller as you increase your distance from the center of
the Earth. For example, if you go to a height that is one earth radius above the surface of
the earth the acceleration of gravity is 2.45 m/s2 (g/4.0). Is the period of a pendulum longer
or shorter?
Ans; Since, the period of a pendulum:
𝑻∝
𝟏
√𝒈
Therefore, as the distance from the center of the Earth increases, the period of a pendulum gets
longer. And, according to given scenario;
L
T′ = 2π√
g′
Where, g’=g/4
L
T′ = 2π√𝑔
⁄4
L
T′ = 2 (2π√ ) = 2𝑇
𝑔
Thus, the period becomes double.
8. The frequency of a pendulum is (1/Period). That is, if the period is the number of second per
cycle the frequency is the number of cycles per second. Another name for cycles per second
is Hertz (Hz). What is the frequency of a pendulum that has a period of 0.5s?
Ans;
𝑓=
1
1
=
= 𝟐𝑯𝒛
𝑇 0.5
Discussion and conclusion
From this lab, it is concluded that a simple pendulum is an effective device to observe oscillatory
motion, and its motion characteristics can be studied experimentally. From readings obtained in
part A and observing the trend of period values against length of the pendulum, it can be easily
seen that with the increase in length of the simple pendulum, time period also increases. It does
not show a complete linear trend although due to some anomalous points, while recording
observations, yet it was observed that, at 0.5 m length of pendulum, the time period is 1.264 s, and
increases at 0.4 m length of pendulum (i.e. 1.4175 s). The opposite happens when the length is
decreased, that is period decreases as the square root of the length. Therefore we can conclude that;
“Time period is directly proportional to the square root of length of simple pendulum”.
Mathematically;
𝑇 ∝ √𝑙 − − − −𝐸𝑞(1)
Where,
T = Time period of simple pendulum
l = length of simple pendulum
Or, in other words;
“Length of simple pendulum is directly proportional to the square of its time period”.
𝐋 ∝ 𝐓𝟐
This relation can be verified from graph drawn. And from the graph of length versus squared time,
the slope of the graph comes out to be 0.2509 m/s2. And using, mathematical correlations,
experimental value of ‘g’ is calculated, which comes out to be 9.93 m/s2 , which lies in close
proximity with predicted value 9.8 m/s2 , with 1.073% error, which is quite less, and we conclude
that our experiment was accurate, with less sources of error.
In part B, position-time behavior of the simple pendulum is analyzed using motion sensor and
PASCO interface, and time period is found from experimental position-time graph (the time taken
by the pendulum to cover a complete oscillation covering one complete back and forth trip
(distance between two consecutive troughs in our case). In this part of experiment, experimental
error in the value of ‘g’ found experimentally (10.13 m/s2) is somewhat large (i.e. ~3.5%), due to
several sources of error described below.
Moreover, the length-squared time graph results in almost a straight line with constant slope,
which verifies the direct relationship between length and square of time period produced as
signified by the time period equation, provided the y-intercept shows some positive value (0.023
s), which occurs due to some error, and is neglected in our experiment.
Finally, it is concluded that Pendulums are used in accelerometers, devices that measure
acceleration. Further, they are used in seismometers, which measure seismic activity caused by
earthquakes and other movements of the Earth. Pendulums are also used in other scientific
instruments.
Sources of Errors
Various sources of errors include;

The simple pendulum may not be at rest initially, i.e. incorporates some initial vibratory
motion.

Parallax error, i.e. extracting resulting deformation by the measuring scale not at right
angles to the spring of pendulum, so values are affected.

Human reaction error of measuring time.

Uncertainty of motion sensor in Part B.

Inaccurate positioning of motion sensor in Part B, and poor connection between sensor and
DataStudio.

Random error, external forces like wind and air resistance affects oscillations of the
pendulum.

Values are taken when pendulum does not completely reached extremes..

Human uncertainty, due to negligence of observer.

Equipment is not calibrated (i.e. position values incorporate deviation from standard
values).
Mitigation Measures

Experiment should be done in isolated environments.

Parallax error should be removed while taking readings (i.e. readings for length of
pendulum must be taken at right angles to the scale)

The apparatus should be tightened.

The string should be inextensible.

The angle should be small.

Motion should be in plane.

Measuring scale and stop watch should be precise.

Properly make connections between sensor and PASCO interface.

Carefully observe readings from stopwatch, i.e. Readings should be taken, when the
pendulum comes to extreme position, and make sure it is also initially at rest.
Therefore, the experiment is a true demonstration of motion of simple pendulum, and it gives a
typical method to observe oscillatory motion of simple pendulum, and effect of length of the
pendulum on its period.