Define an ideal simple pendulum. Show that motion of a simple pendulum under certain conditions is simple harmonic. Obtain an expression for its period.
A solid sphere of diameter 50 cm and mass 25 kg rotates about an axis through its centre. Calculate its moment of inertia. If its angular velocity changes from 2 rad/s to 12 rad/s in 5 seconds. calculate the torque applied.

Explanation

### Step-by-Step Solution

#### Part 1: Ideal Simple Pendulum and Simple Harmonic Motion

1. **Definition of an Ideal Simple Pendulum**:
An ideal simple pendulum consists of a massless string of length \( L \) attached to a rigid support, with a point mass \( M \) (the bob) at the other end. The motion occurs in a vertical plane, and we assume no air resistance or friction at the pivot.

2. **Conditions for Simple Harmonic Motion (SHM)**:
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