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Introduction

A simple pendulum is formed by a small bob of mass m suspended from a fixed point by a light, inextensible string of length L. When displaced from equilibrium and released, the bob oscillates under the influence of gravity.

T = 2\pi \sqrt{\tfrac{L}{g}}

Here, T is the time period of one oscillation, L is the length of the pendulum, and g is the acceleration due to gravity.

Forces Involved

Gravitational Force

The bob experiences its weight acting vertically downward.

F_{g} = m g

Here, Fg is the gravitational force (weight), m is the mass of the bob, and g is the acceleration due to gravity.

Tension

The string provides a tension force directed along its length, balancing part of the weight and keeping the bob constrained to move along a circular arc.

T - m g \cos\theta = m \tfrac{v^2}{L}

Here, T is the tension in the string, m is the bob’s mass, g is acceleration due to gravity, θ is the angular displacement, v is the bob’s velocity, and L is the string length.

Restoring Force

The tangential component of gravity acts as the restoring force, pulling the bob back to its mean position.

F_{restoring} = - m g \sin\theta

Here, Frestoring is the restoring force, m is the mass of the bob, g is acceleration due to gravity, and θ is the angular displacement.

Equation of Motion

Applying Newton’s second law along the arc gives the differential equation of motion.

m L \tfrac{d^2\theta}{dt^2} = - m g \sin\theta

Here, m is the bob’s mass, L is the string length, θ is angular displacement, t is time, and g is acceleration due to gravity.

\tfrac{d^2\theta}{dt^2} + \tfrac{g}{L} \sin\theta = 0

This is the general differential equation for pendulum motion, where θ is angular displacement, g is acceleration due to gravity, and L is the length of the pendulum.

Small Angle Approximation

For small angles (θ < 10°), we approximate sinθ ≈ θ (in radians). This simplifies the motion to simple harmonic motion (SHM).

\tfrac{d^2\theta}{dt^2} + \tfrac{g}{L}\,\theta = 0

Here, θ is angular displacement, g is gravitational acceleration, and L is string length. This equation represents SHM.

The solution describes oscillations with angular frequency ω = √(g/L).

T = 2\pi \sqrt{\tfrac{L}{g}}

Here, T is the time period, L is the string length, and g is the acceleration due to gravity.

Energy Analysis

The pendulum continuously exchanges energy between kinetic and potential forms while total mechanical energy remains constant (ignoring air resistance).

E = K + U = \tfrac{1}{2} m v^2 + m g h

Here, E is total energy, K is kinetic energy, U is potential energy, m is the bob’s mass, v is velocity, g is gravitational acceleration, and h is height relative to the mean position.

At the mean position, energy is entirely kinetic; at extreme positions, it is entirely potential.

Measurements and Observations

The period is independent of the bob's mass.
For small oscillations, the time period depends only on length L and gravity g.
Longer pendulums have longer periods.
For large angles, the motion deviates from SHM and the period increases slightly.

Constants and Recommended Values

Constant	Meaning	Suggested value
g	Gravitational acceleration	9.8 m/s²
L	Length of pendulum	0.5 – 2.0 m
θ	Initial displacement angle	< 10° for SHM

6 - Simple Pendulum