Simple Harmonic Motion:
Simple harmonic motion (SHM) is a type of periodic motion where an object oscillates back and forth around an equilibrium position. The motion is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement.

Oscillations of a Simple Pendulum:
- A simple pendulum consists of a mass (bob) suspended from a fixed point by a light string.
- When the pendulum is displaced from its equilibrium position and released, it oscillates back and forth.
- The motion of a simple pendulum is approximately simple harmonic for small angles of displacement (less than 15 degrees).

Verification of Simple Harmonic Motion:
The restoring force in a simple pendulum is given by the formula: F = -mg sinθ, where m is the mass of the bob, g is the acceleration due to gravity, and θ is the angle of displacement. For small angles, sinθ ≈ θ in radians.
- Substituting sinθ ≈ θ into the formula, we get: F = -mgθ.
- This is in the form of Hooke's Law (F = -kx), where k is the spring constant and x is the displacement. Therefore, the oscillations of a simple pendulum are simple harmonic.

Time Period of Oscillations:
The time period of oscillations for a simple pendulum can be calculated using the formula: T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity (approximately 9.81 m/s²).
- For example, if the length of the pendulum is 1 meter, the time period of oscillations would be: T = 2π√(1/9.81) ≈ 2.01 seconds.
In conclusion, the oscillations of a simple pendulum exhibit simple harmonic motion for small angles of displacement, and the time period of oscillations can be calculated using the formula T = 2π√(l/g).