A load of mass m falls from a height h on the scale pan hung from a spring as shown. If the spring constant is k and the mass of the scale pan is zero and the mass m does not bounce relative to the pan, then the amplitude of vibration is

Step-by-step solution

From the energy conservation principle, m g h = 1 2 k X 0 2 - m g X 0 where, X 0 is maximum elongation in spring. ⇒   1 2 k X 0 2 - m g X 0 - m g h = 0 ⇒   X 0 2 - 2 m g k X 0 - 2 m g k h = 0 X 0 = 2 m g 2 ± 2 m g k 2 + 4 × 2 m g k h 2 = m g 2 + m g k 2 + 2 m g k h Amplitude = elongation in spring for lowest extreme position X 0 - elongation in spring for equilibrium position X 1 = X 0 - X 1 = m g k 1 + 2 h k m g   , ∵ X 1 = m g k