A mass 𝑀, attached to a horizontalspring, executes SHM with a amplitude𝐴1. When the mass 𝑀 passes through itsmean position then a smaller mass 𝑚 isplaced over it and both of them movetogether with amplitude 𝐴2. The ratio of(𝐴1𝐴2) is
Question
A mass 𝑀, attached to a horizontalspring, executes SHM with a amplitude𝐴1. When the mass 𝑀 passes through itsmean position then a smaller mass 𝑚 isplaced over it and both of them movetogether with amplitude 𝐴2. The ratio of(𝐴1𝐴2) is
Solution
The amplitude of a simple harmonic motion (SHM) is directly proportional to the energy of the system.
In the first case, the energy of the system is given by (1/2)Mω²A₁², where ω is the angular frequency of the oscillation.
In the second case, when the smaller mass m is added, the total mass becomes (M+m) and the energy of the system becomes (1/2)(M+m)ω²A₂².
Since the energy of the system must be conserved, we can equate the two expressions for energy:
(1/2)Mω²A₁² = (1/2)(M+m)ω²A₂²
Solving this equation for the ratio A₁/A₂ gives:
A₁/A₂ = sqrt((M+m)/M)
So, the ratio of the amplitudes A₁ to A₂ is the square root of the ratio of the total mass (M+m) to the original mass M.
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