A mass m₁ performs S.H.M. with amplitude A which is connected to horizontal spring. While mass m₁ passing through mean position, another mass m₂ ( m₂ < m₁ ) is placed on it so that both masses move together with amplitude A₁ . The ratio of A/A₁ is

Step-by-step solution

Velocity of mass m₁ at the mean position is given by v = A = A k m₁ . When mass m₂ is placed on m₁ , the horizontal momentum of the system is conserved. m₁ v = (m₁ + m₂)v₁ v₁ = m₁ v m₁ + m₂ The new angular frequency of the system is ₁ = k m₁ + m₂ . The new amplitude A₁ is related to the new maximum velocity v₁ by v₁ = A₁ ₁ . Substituting the values of v₁ and ₁ : A₁ k m₁ + m₂ = m₁ m₁ + m₂ A k m₁ A₁ = A m₁ m₁ + m₂ m₁ + m₂ m₁ A₁ = A m₁ m₁ + m₂ Rearranging to find the ratio A A₁ : A A₁ = [ m₁ + m₂ m₁ ]^ 1 2 Answer: [ m