INTRODUCTION

A simple pendulum can be described as a device where its point mass is attached to a
light in-extensible string and suspended from a fixed support. The vertical line passing
through thefixed support is the mean position of a simple pendulum. The vertical distance
between the point of suspension and the centre of mass of the suspended body, when it is
in the mean position, is called the length of the simple pendulum, denoted by L. This form
of the pendulum is based on the resonant system having a single resonant frequency.

Simple Pendulum Definition

A simple pendulum is a mechanical arrangement that demonstrates periodic motion. The

simple pendulum comprises a small bob of mass ‘m’ suspended by a thin string secured
to aplatform at its upper end of length L.

The simple pendulum is a mechanical system that sways or moves in an oscillatory

motion.This motion occurs in a vertical plane and is mainly driven by gravitational force.
Interestingly, the bob that is suspended at the end of a thread is very light; somewhat, we can
say it is even mass less. The period of a simple pendulum can be made extended by
increasingthe length string while taking the measurements from the point of suspension to
the middle ofthe bob. However, it should be noted that if the mass of the bob is changed,
the period will remain unchanged. The period is influenced mainly by the position of the
pendulum in relation to Earth, as the strength of the gravitational field is not uniform
everywhere.

In addition, pendulums are a common system whose usage is seen in various instances.
Someare used in clocks to keep track of the time, while some are just used for fun in case
of a child’s swing. In some cases, it is used in an unconventional manner, such as a sinker
on a fishing line. In any case, we will explore and learn more about the simple pendulum
on this page. We will discover the conditions under which it performs simple harmonic
motion as well as derive an interesting expression for its period.

Important Terms

• The oscillatory motion of a simple pendulum: Oscillatory motion is defined as

theto and fro motion of the pendulum in a periodic fashion, and the centre point
of oscillation is known as the equilibrium position.
• The time period of a simple pendulum: It is defined as the time taken by
thependulum to finish one full oscillation and is denoted by “T”.
• The amplitude of a simple pendulum: It is defined as the distance travelled by
thependulum from the equilibrium position to one side.

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 • Length of a simple pendulum: It is defined as the distance between the point of
suspension to the centre of the bob and is denoted by “l”.

Time Period of Simple Pendulum

A point mass M is suspended from the end of a light in extensible string, whose upper end is
fixed to a rigid support. The mass is displaced from its mean position.

Assumptions:

• There is negligible friction from the air and the system

• The arm of the pendulum does not bend or compress and is mass less
• The pendulum swings in a perfect plane
• Gravity remains constant.

Time Period of Simple Pendulum Derivation

Using the equation of motion, T – mg cosθ = mv2L

The torque tends to bring the mass to its equilibrium position, τ

= mgL × sinθ = mgsinθ × L = I × α

For small angles of oscillations sin θ ≈ θ,

Therefore, Iα = -mgLθ α = -(mgLθ)/I

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– ω02 θ = -(mgLθ)/I ω02

= (mgL)/I

ω0 = √(mgL/I)

Using I = ML2, [where I denote the moment of inertia of bob] we

get, ω0 = √(g/L)

Therefore, the time period of a simple pendulum is given by,

T = 2π/ω0 = 2π × √(L/g)

Energy of Simple Pendulum

Potential Energy

The potential energy is given by the basic equation

Potential energy = mgh m is the mass of the object

g is the acceleration due to gravity h is the height of

the object

However, the movement of the pendulum is not free fall; it is constrained by the rod or string.
The height is written in terms of angle θ and length L. Thus, h = L(1 – cos θ)

When θ = 90°, the pendulum is at the highest point. Then cos 90° = 0, and h =

L.Therefore,

Potential Energy = mgL

When θ = 0°, the pendulum is at the lowest point. Then, cos 0° = 1. Therefore h = L (1-1)

=0Potential energy = mgL (1-1) = 0

At all the points in between the potential energy is given as mgL (1 – cos θ).

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Kinetic Energy

The kinetic energy of the pendulum is given as K.E = (1/2) mv2 m

is the mass of the pendulum v is the velocity of the pendulum At

the highest point, the kinetic energy is zero, and it is maximum at

the lowest point. However, the total energy as a function of time

is constant.

Mechanical Energy of the Bob

In a simple pendulum, the mechanical energy of simple pendulum is conserved. E =

KE + PE= 1/2 mv2 + mgL (1 – cos θ) = constant

⇒ Note:

• If the temperature of a system changes, then the time period of the simple pendulum
changes due to a change in the length of the pendulum.
• A simple pendulum is placed in a non-inertial frame of reference (accelerated lift,
horizontally accelerated vehicle, vehicle moving along an inclined plane).

The mean position of the pendulum may change. In these cases, g is replaced by “g effective”
for determining the time period (T).

For example,

• A lift moving upwards with acceleration ‘a’, then, T = 2π × √(L/geff) = 2π √[L/(g + a)]
• If the lift is moving downward with acceleration ‘a’, then T = 2π × √(L/geff) = 2π
√[L/(g – a)]
• For a simple pendulum of length L is equal to the radius of the earth ‘R’, L = R = 6.4
x 106 m, then the time period T = 2π √R/2g
• For infinitely long pendulum L > > R near the earth surface, T = 2π × √(R/g)

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Physical Pendulum

A simple pendulum is an idealised model. It is not achievable in reality. But the physical
pendulum is a real pendulum in which a body of finite shape oscillates. From its frequency of
oscillation, we can calculate the moment of inertia of the body about the axis of rotation.

Consider a body of irregular shape and mass (m) that is free to oscillate in a vertical plane
about a horizontal axis passing through a point, and weight mg acts downward at the centre of
gravity (G).

If the body is displaced through a small angle (θ) and released from this position, a torque is

exerted by the weight of the body to restore its equilibrium. τ = -mg × (d sinθ) τ = I α

I α = – mgdsinθ

I . d2θ/dt2 = – mgdsinθ

Where I = moment of inertia of a body about the axis of rotation. d2θ/dt2

= (mgd/I) θ [Since, sinθ ≈ θ] ω0 = √[mgd/I].

Time Period of Physical Pendulum

T = 2π/ω0 = 2π × √[I/mgd]

For ‘I’, applying the parallel axis theorem,

I = Icm + md2

Therefore, the time period of a physical pendulum is given by, T = 2π × √[(Icm + md2)/mgd]

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 EXPERIMENT

Aim:

Using a simple pendulum, plot L-T and L-T2 graphs and use it to find the effective length of
the second’s pendulum.

Apparatus and Material Required:

• Clamp stand
• Split Cork
• Heavy metallic spherical bob with a hook
• Long and strong cotton thread
• Stopwatch
• Meter scale
• Graph Paper
• Pencil Eraser

Theory:

The simple pendulum exhibits Simple Harmonic Motion (SHM) as the acceleration of the
pendulum bob is directly proportional to the displacement from the mean position and is
always directed towards it. The time period (T) of a simple pendulum for oscillations of small
amplitude is given by the relation

T=2π √L/G
Where L is the length of the pendulum and g is the acceleration of gravity

Procedure:

1. Place the clamp stand on the table. Tie the hook attached to the pendulum bob, to one
end of the string of about 150 cm in length and the other end of the string through two
half-pieces of a split cork.
2. Clamp the split cork firmly to the clamp stand such that the line of separation between
the two pieces of the split cork is at right angles to the line OA along which the
pendulum oscillates as given in the figure. Mark the edge of the table a vertical line
parallel to and just behind the vertical thread OA, the position of the bob at rest. Take
care that the bob hangs vertically (about 2 cm above the floor) beyond the edge of the
table so that it is free to oscillate.
3. Measure the effective length of the simple pendulum as shown in the figure.

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 Setup of dissipation of energy of an Effective length of simple
oscillating simple pendulum pendulum

1. Displace the bob not more than 15 degrees from the vertical position OA and then
gently release it. If you notice the stand to be shaky, put a heavy object on its base.
Make sure that the bob oscillates in a vertical plane about its rest and does not (i) spin
about its own axis (ii) move up and down while oscillating (iii)revolve in an elliptic
path around its mean position.
2. Keep the pendulum oscillating for a few minutes. After the completion of few
oscillations, start the stopwatch as the thread attached to the bob crosses the mean
position. Consider it as a zero oscillation.
3. Keep counting the oscillation 1,2,3…n every time the bob crosses the mean position.
Stop the stopwatch at the count of n oscillations. For better results, n should be chosen
such that the time taken to complete n oscillations is 50 s or more. Read the total time
taken for n oscillations. Repeat the observation a few times by noting down the time
for the same n number of oscillations. Once noted down, take the mean of the
readings. Calculate the time for one oscillation, i.e., the time period T ( = t/n) of the
pendulum.
4. Change the length of the pendulum, by about 10 cm. Repeat step 6 again for finding
the time (t) for about 20 oscillations or more for the new length and find the mean
time period. Take 5 or 6 more observations for different lengths of the pendulum and
find the mean time period in each case.
5. Report observations in tabular form with proper units and significant figures.
6. Take effective length L along the x-axis and T2 (or T) along the y-axis, using the
observed values from the table. Choose suitable scales on these axes to represent L
and T2 (or T). Plot a graph between L and T2 as shown in figure 2 and also between L
and T as shown in figure 1.

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Observation:

The radius of the pendulum of the bob = …. cm

Length of the hook = ….. cm

Least count of the meter scale = ….. mm

Least count of the stopwatch = ….. s

S.No Length of the Number of Time for n oscillations t(s) Time period T2
pendulam,l (cm) oscillations, n T(=t/n)

(i) (ii) Mean t(s)

1. 50 10 14.6 14.5 14.5 1.45 2.10

2. 60 10 14.9 15.4 15.1 1.51 2.28

3. 70 10 16 16.6 16.3 1.63 2.65

4. 80 10 18.1 17.8 17.9 1.79 3.20

5. 90 10 18.6 18.9 18.7 1.87 3.49

Plotting Graph (i) L vs T Graph

Plot a graph between L versus T from observations recorded in the table above, taking L
along x-axis and T along the y-axis. You will find that this graph is a curve, which is part of a
parabola as shown in Figure 1.

(ii) L vs T2 Graph

Plot a graph between L versus T2 from observations recorded in the table, taking L along the
x-axis and T2 along the y-axis. You will find that the graph is a straight line passing through
the origin as shown in figure 2.

(iii) From the L versus T2 graph, determine the effective length of the second’s pendulum for
T2 = 4s2.

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 Result:

The graph L versus T is curved, convex upwards.

The graph L versus T2 is a straight line.

The effective length of the second’s pendulum from the L versus T2 graph is cm.

Precautions
1. Use a good stop watch of small least count.
2. Thread should be in extensible and of neglibible mass.
3. Point of suspension should be attached to a rigid support.
4. The bob should not spin during vibration.
5. Amplitude of the oscillations should be small.

Sources of Error
1. Point of suspension may not be rigid.
2. The amplitude of oscillation is very large.
3. The bob spins during oscillation.
4. There may be delay in starting and stopping of stop watch.
5. The air currents may disturb vibration.

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 FUTURE SCOPE OF THE PROJECT

 Enhanced Understanding of Oscillatory Motion: The experiment can be extended to study

damping effects, changes in oscillation amplitude, or influence of environmental factors like air
resistance.

 Application in Timekeeping: Insights from the pendulum's motion can be applied to develop or
calibrate mechanical clocks.

 Testing Local Gravitational Acceleration: Use the T2∝L relationship to calculate g at different
locations.

 Advanced Studies: Incorporate modern sensors and software for more precise measurements of
TTT and LLL, leading to reduced experimental error.

 Interdisciplinary Applications: Explore the pendulum's role in seismology, engineering (vibration

analysis), or architecture (pendulum-based damping systems).

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 BIBLIOGRAPHY

1. www.google.com
2. www.physicsprojects.com
3. www.wikipedia.org
4. Comprehensive practical book

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