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Question
The string the spring and the pulley shown in figure are light. Find the time period of the mass m.
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Solution
Let l be the extension in the spring when mass m is hung.
Let T1 be the tension in the string; its value is given by,
T1 = kl = mg
Let x be the extension in the string on applying a force F.
Then, the new value of tension T2 is given by,
T2 = k(x + l)
Driving force is the difference between tensions T1 and T2.
∴ Driving force = T2 − T1 = k(x + l) − kl
= kx
\[\text{Acceleration}, a = \frac{kx}{m}\]
\[\text { Time period } \left( T \right)\] is given by,
\[T = 2\pi\sqrt{\frac{\text { displacement }}{\text { Acceleration }}}\]
\[ = 2\pi\sqrt{\frac{x}{kx/m}} = 2\pi\sqrt{\frac{m}{k}}\]
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