A pendulum is oscillating with an angular amplitude of 90 ° as shown in the figure. The value of angle α , made by the string with the vertical, at which the acceleration is directed horizontally is

Step-by-step solution

For horizontal acceleration, net force in vertical direction should be zero. Loss in potential energy = gain in kinetic energy m g h = 1 2 m v 2 m g l cos ⁡ α = 1 2 m v 2 ∴   v = 2 g l cos ⁡ α ∵ The net force towards O is T - m g cos ⁡ α = m v 2 l = m 2 g l cos ⁡ α   l = 2   m g cos ⁡ α ∴ T = 3 m g cos ⁡ α ∴ T cos ⁡ α = m g Or 3 m g   c o s 2   α = m g ∴ α = c o s - 1 1 3