Pendulum Motion and Simple
Pendulum
Name:Ayushmann
Class:11-A
Roll No.:1113A
Introduction
Pendulum motion refers to the oscillatory movement of an object that swings back and
forth about a fixed point under the influence of gravity. This simple mechanical system
is a key concept in physics, specifically in the study of oscillations and harmonic motion.
A pendulum consists of a mass (called the bob) attached to a string or rod, which is fixed
at one point, allowing the bob to swing in an arc. The motion of a pendulum is governed
by simple harmonic motion (SHM), a type of repetitive movement that occurs when a
restoring force is proportional to the displacement from the equilibrium position.
What is a Pendulum Motion?
A simple pendulum consists of a small mass (the bob) attached to the end of a
lightweight, inextensible string or rod of fixed length, with the other end attached to a
pivot point. When displaced from its equilibrium position (i.e., when the bob is moved to
one side), the pendulum experiences a restoring force due to gravity. This force acts in
such a way that it brings the bob back toward the equilibrium position, causing the
pendulum to oscillate.
The oscillations of the pendulum are periodic, meaning they repeat after a fixed interval
of time. The time it takes to complete one full oscillation (from one extreme point to the
other and back) is called the period of the pendulum. The motion of the pendulum can
be analyzed in terms of both displacement and time.
Key terms in Pendulum Motion
1. Amplitude(A): The maximum displacement from the equilibrium position.
2. Period (T): The time taken for one complete oscillation.
3. Frequency (f): The number of oscillations per second, the reciprocal of the period
(f=1/T).
4. Restoring Force: The force that acts to return the bob to its equilibrium position,
which is due to gravity.
5. Equilibrium Position: The position where the bob would rest if there were no
motion (typically at the lowest point of the swing).
Deriving the equation of motion
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
– ω2θ = -(mgLθ)/I
ω2 = (mgL)/I
ω = √(mgL/I)
Using I = ML2, [where I denote the moment of inertia of bob]
we get, ω = √(g/L)
Therefore, the time period of a simple pendulum is given by,
T = 2π/ω0 = 2π × √(L/g)\
Factors Affecting Pendulum Motion
While the period of a simple pendulum is primarily influenced by the length of
the string and gravitational acceleration, there are several factors that can affect
the motion of a real-world pendulum:
1. Length of the Pendulum: As derived in the equation of motion, the period
of the pendulum is directly proportional to the square root of its length. A
longer pendulum takes more time to complete one oscillation than a
shorter one.
2. Acceleration due to Gravity: The value of g depends on the location of the
pendulum. At higher altitudes, where the gravitational force is slightly
weaker, the period of the pendulum will be longer.
3. Amplitude: For small angles of displacement (typically less than 15°), the
period is independent of the amplitude, making the motion approximately
simple harmonic. For larger amplitudes, the approximation sinθ≈θ breaks
down, and the period increases slightly with amplitude.
4. Air Resistance and Damping: In a real-world scenario, air resistance acts
on the pendulum, gradually converting mechanical energy into heat and
slowing the motion over time. This results in damping, which causes the
amplitude of oscillation to decrease, and the pendulum eventually comes
to rest.
5. Mass of the Bob: In an ideal simple pendulum, the mass of the bob does
not affect the period. However, in practice, factors like the shape and air
resistance can make the mass slightly influential.
Types of Pendulums
There are different types of pendulums, each with its specific characteristics
and behavior:
1. Simple Pendulum: As discussed earlier, this consists of a point mass
attached to a string or rod of negligible mass.
2. Physical Pendulum: A physical pendulum is a rigid body swinging about a
horizontal axis. Unlike a simple pendulum, the mass of the object cannot
be neglected, and the moment of inertia must be considered when
calculating its period. For a physical pendulum, the period is given by:
5. Applications of Pendulum Motion
Pendulum motion has several practical applications:
1. Clocks: The pendulum is one of the most important components of
mechanical clocks. The regular oscillations of the pendulum help to keep
time with high accuracy. The period of the pendulum is adjusted to be
constant, which makes it a reliable timekeeping mechanism.
2. Seismographs: Pendulums are used in seismographs to measure ground
movements during an earthquake. The motion of the pendulum is
translated into a recording that indicates the magnitude and intensity of
the seismic waves.
3. Accelerometers: Pendulum-based systems are used in accelerometers to
measure changes in velocity. These are commonly used in devices such as
smartphones and automotive safety systems (e.g., airbags).
