**Derivation of Simple Pendulum in SHM**

A simple pendulum consists of a mass (bob) attached to a string or rod of negligible mass and fixed at a pivot point. When the pendulum is displaced from its equilibrium position and released, it oscillates back and forth. The motion of a simple pendulum can be described by Simple Harmonic Motion (SHM). Let's derive the equation of motion for a simple pendulum in SHM.

**Assumptions:**
1. The string or rod is massless and inextensible.
2. The angle of displacement is small, so we can use the small-angle approximation.

**Derivation:**

1. **Defining Variables:**
- Let 'm' be the mass of the bob.
- Let 'L' be the length of the string or rod.
- Let 'θ' be the angular displacement of the pendulum from the vertical equilibrium position.
- Let 'g' be the acceleration due to gravity.

2. **Forces Acting on the Pendulum:**
- The weight of the bob acts vertically downwards and can be decomposed into two components: mg sin(θ) along the radial direction and mg cos(θ) along the tangential direction.
- The tension in the string or rod acts along the tangential direction towards the pivot point.
- The net force acting on the bob is the sum of the tangential components: T - mg cos(θ), where 'T' is the tension in the string or rod.

3. **Angular Acceleration:**
- The torque acting on the bob is given by the product of the force and the perpendicular distance from the pivot point, which is 'L' in this case.
- The torque can be written as: τ = (T - mg cos(θ)) * L.
- According to Newton's second law for rotational motion, τ = Iα, where 'I' is the moment of inertia and 'α' is the angular acceleration.
- For a simple pendulum, the moment of inertia is given by I = mL².
- Substituting the values, we get: (T - mg cos(θ)) * L = mL² * α.

4. **Restoring Force and SHM:**
- The restoring force is the component of the net force that acts towards the equilibrium position.
- The restoring force can be written as: F = -mg sin(θ).
- Using the small-angle approximation sin(θ) ≈ θ for small θ, we have F ≈ -mgθ.
- From the equation of angular acceleration, we can substitute τ = Iα as L * F = mL² * α.
- Simplifying the equation, we get: -mgθ = mL² * α.

5. **Equation of Motion:**
- Dividing both sides by mL², we obtain: -gθ/L = α.
- The angular acceleration α is the second derivative of θ with respect to time, so we can write: -g/L * θ = d²θ/dt².
- This equation represents the equation of motion for a simple pendulum in SHM.

Hence, the equation of motion for a simple pendulum in SHM is given by:
d²θ/dt² + (g/L) * θ