Ncert Exercises & Solutions

A mass \(M\) is attached to a spring system as shown in the figure. If the mass is displaced from its equilibrium position and then released, what is the time period of its oscillation?

1. \(2\pi \sqrt{\dfrac{M}{k}} \)
2. \(2\pi \sqrt{\dfrac{M}{2k}} \)
3. \(2\pi \sqrt{\dfrac{M}{4k}} \)
4. \(2\pi \sqrt{\dfrac{2M}{3k}} \)

Hint: \(F=-kx\)

Step 1: Find the tension in the lower string.
\(\dfrac{T}{2}=k(2x)\)
\(\Rightarrow T=4kx\)
Step 2: Find the angular frequency of spring-mass system.
\(ma=-4kx\)
\(\Rightarrow a=-\dfrac{4kx}{m}\)
\(\Rightarrow -\omega^2x=-\dfrac{4kx}{m}\)
\(\Rightarrow \omega=\sqrt{\dfrac{4k}{m}} \)
Step 3: Find the time period of spring-mass system.
\(\omega=\dfrac{2\pi }{T}\)
\(\Rightarrow \sqrt{\dfrac{4k}{M}}=\dfrac{2\pi}{T}\)
\(\Rightarrow T=2\pi \sqrt{\dfrac{M}{4k}} \)
Hence, option (3) is the correct answer.

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