An iron ring of mass m1 oscillates in it's an plane with time period of π sec. Assume that the amplitude of oscillation is small. When it is passing through the mean position, a magnet of mass m is attached to the lowest point. If the amplitude of oscillation decreases by factor of 7, find .

Engineering

Physics

Simple Harmonic Motion

Question

An iron ring of mass m₁ oscillates in it's an plane with time period of π sec. Assume that the amplitude of oscillation is small. When it is passing through the mean position, a magnet of mass m is attached to the lowest point. If the amplitude of oscillation decreases by factor of 7, find $\frac{m}{m_{1}}$ .

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$Ω_{0} = Q_{0} ω_{0} = Q_{0} \times \frac{2 π}{π}$

$ω_{0} = 2 = \sqrt{\frac{2 m_{1} R^{2}}{m_{1} gR}} = \sqrt{\frac{2 R}{g}}$

L_i = L_f

$(m_{1} R^{2} + m_{1} R^{2}) Ω_{0} = (m_{1} R^{2} + m_{1} R^{2} + m (2 R)^{2}) \times Ω$

$Ω = θ \times ω^{'}$

$ω^{'} = \sqrt{\frac{I}{mgd}} = \sqrt{\frac{2 m_{1} R^{2} + 4 {mRR}^{2}}{g (m_{1} R + m \times 2 R)}} = \sqrt{\frac{2 R}{g}} = 2$

$2 m_{1} R^{2} \times 2 θ_{0} = (2 m_{1} + 4 m) R^{2} \times 2 \times θ_{0}^{'}$

$m_{1} θ_{0} = (m_{1} + 2 m) \frac{θ_{0}}{7}$

m = 3m₁

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