Question

A block of mass M is kept on a smooth surface and touches the two springs as shown in the figure but not attached to the springs. Initially springs are in their natural length. Now, the block is shifted $\left(l_0/2\right)$ from the given position in such a way that it compresses a spring and released. The time-period of oscillation of mass will be:

• A. $\frac{\pi }{2}\sqrt{\frac{M}{k}}$
• B. $2\pi \sqrt{\frac{M}{5k}}$
• C. $\frac{3\pi }{2}\sqrt{\frac{M}{k}}$
• D. $\pi \sqrt{\frac{M}{2k}}$

Answer 1

Answer: (C)

$\frac{3\pi }{2}\sqrt{\frac{M}{k}}$

Half the oscillation is completed with one spring and the other half with the other spring. Hence,
$T=\frac{T_1}{2}+\frac{T_2}{2}$

$=\frac{2\pi \sqrt{\frac{M}{k}}}{2}+\frac{2\pi \sqrt{\frac{M}{4K}}}{2}$
$=\pi \sqrt{\frac{M}{K}}[1+\frac{1}{2}]$
$=\frac{3\pi }{2}\sqrt{\frac{M}{K}}$