Simple Pendulum: Physics and Motion Analysis

Simple Pendulum Explained

A simple pendulum ideally consists of a point mass, m, suspended by a massless, inextensible rope of length L. The upper end of the rope is fixed, and the pendulum oscillates in a vacuum, free from friction forces.

Pendulum Motion

If the mass is displaced from its equilibrium position (point A), the pendulum swings in a vertical plane, exhibiting periodic motion. When the pendulum mass reaches a point B, its weight (mg) can be resolved into two components:

• mg cos(α): This component is balanced by the tension in the rope.
• -mg sin(α): This is the restoring force (F) that tends to bring the pendulum back to its equilibrium position.

The restoring force F is proportional to sin(α). Therefore, the resulting motion is generally not simple harmonic motion (SHM). The motion of a simple pendulum approximates SHM only when the restoring force is linearly proportional to the displacement. This occurs when the angle of oscillation (α) is small enough that sin(α) ≈ α (where α is in radians).

Period of Oscillation

The period (T) is the time taken for the pendulum to complete one full cycle (oscillation). For small oscillations (where SHM is approximated), the period is given by the expression:

T = 2π√(m / k)

Under the conditions for a simple pendulum's motion, the effective spring constant k is equivalent to (mg) / L. Substituting this, the formula becomes:

T = 2π√(L / g)

This shows that the period of a simple pendulum, for small angles, is:

• Independent of the suspended mass (m).
• Directly proportional to the square root of its length (L).
• Inversely proportional to the square root of the acceleration due to gravity (g).

The frequency (f) of the motion is the inverse of the period (f = 1/T) and represents the number of oscillations the particle completes per unit time.

Experiment Objectives

• To study the motion of a simple pendulum as an example of simple harmonic motion.
• To determine the value of the acceleration due to gravity (g).
• To analyze the factors influencing the period, specifically studying how the period depends on the angle of oscillation, the oscillating mass, and the length of the pendulum.

Instruments and Equipment Used

• Tape Measure: Accuracy ± 0.1 cm, Range up to 300.0 cm
• Vernier Caliper: Accuracy ± 0.005 cm, Range up to 15.250 cm
• Balance: Accuracy ± 0.0001 g, Range up to 160.0000 g
• Stopwatch: Accuracy ± 0.01 s, Range up to 999.99 s (assuming '9th 595,999' meant 999.99s)
• Spheres: Made of different materials (e.g., metal and wood)
• Support Stand & Rope