Physics Complete: Simple Pendulum and SHM

A point sized heavy mass suspended with a inextensible string from a rigid support is called simple pendulum. For less angle of oscillation it executes simple harmonic motion.

The forces acting on the bob are the force T, tension in the string and the gravitational force Fg (= m g), as shown in figure. The string makes an angle θ with the vertical. We resolve the force Fg into a radial component Fg cos θ and a tangential component Fg sin θ.

The radial component is canceled by the tension, since there is no motion along the length of the string. The tangential component produces a restoring torque about the pendulum’s pivot point. This torque always acts opposite to the displacement of the bob so as to bring it back towards its central location. The central location is called the equilibrium position ( θ = 0), because at this position the pendulum would be at rest if it were not swinging.

The restoring torque τ is τ = –L (Fg sin θ) where the negative sign indicates that the torque acts to reduce θ, and L is the length of the moment arm of the force Fg sin θ about the pivot point. For rotational motion , τ = I α where I is the pendulum’s rotational inertia about the pivot point and α is its angular acceleration about that point.