A body of mass m is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass m is slightly pulled down and released, it oscillates with a time period of 2 s. When the mass m is increased by 1 kg, the time period of oscillations becomes 4 s. The value of m (in kg) is

Time period,

$T = 2 π \sqrt{\frac{m}{k}}$

Case I $T_{1} = 2 π \sqrt{\frac{m}{k}}$ …(i)

Case II When the mass m increased by 1 kg, then net mass = m + 1

$\Rightarrow T_{2} = 2 π \sqrt{\frac{m + 1}{k}}$ …(ii)

On dividing Eq. (ii) by Eq. (i), we get

$\frac{T_{2}}{T_{1}} = \sqrt{\frac{m + 1}{m}} \Rightarrow \frac{4}{2} = \sqrt{\frac{m + 1}{m}}$

$\Rightarrow 4 = \frac{m + 1}{m} \Rightarrow 4 = 1 + \frac{1}{m}$

$\Rightarrow \frac{1}{m} = 3$

$\Rightarrow m = \frac{1}{3} kg$

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