Question

A mass $M$ is suspended from a light spring. An additional mass $m$ added displaces the spring further by a distance $X$. Now the combined mass will oscillate on the spring with period

• A. $T= 2\pi \sqrt{\frac{mg}{X\left(M+m\right)}}$
• B. $T= 2\pi \sqrt{\frac{X\left(M+m\right)}{mg}}$
• C. $T= \frac{\pi}{2} \sqrt{\frac{X\left(M+m\right)}{mg}}$
• D. $T= 2\pi \sqrt{\frac{\left(M+m\right)}{mg}}$

Answer 1

Answer: (B)

Time period of spring block system is given by,
$T= 2\pi \sqrt{\frac{mass \,of \, block }{spring \, constant}}$
Here, mass of block = $(M + m)$
Spring constant $k = \frac{mg}{X} $
$[ \because \:\: At \, equilibrium, (M +m)g = k (X_0 + X ) \, or , mg = kX ( Initially, Mg = kX_0)]$
$ \therefore \:\:\: T= 2\pi \sqrt{\frac{\left(M +m\right)}{\frac{mg}{X}}} = 2\pi \sqrt{\frac{\left(M + m\right)X}{mg}}$