A simple pendulum consists of a bob of mass m suspended by a light string of length L . It is given an initial angular displacement ₀ and released, oscillating with a total mechanical energy E₁ . Later, the bob is replaced by a heavier one of mass 2m , and the pendulum is again released from the same initial angular displacement ₀ , oscillating with a total mechanical energy E₂ . The ratio E₂ E₁ is:

Step-by-step solution

For a simple pendulum of length L and maximum angular displacement ₀ , the maximum height reached by the bob from its lowest position is h = L(1 - ₀) . The total mechanical energy of the pendulum is equal to its maximum potential energy: E = mgh = mgL(1 - ₀) Since L and ₀ are kept constant, the total mechanical energy is directly proportional to the mass of the bob ( E m ). When the mass is doubled from m to 2m , the new total mechanical energy E₂ becomes twice the initial energy E₁ . Therefore, E₂ E₁ = 2 . Answer: