Question

All the surfaces shown in the figure are frictionless. The mass of the inclined wedge is $M$, that of the block is $m$ and spring constant of the spring is $k$. Initially, the block and wedge are at rest. The spring is stretched by a small length and then released. Find the time period of oscillation of the system.

• A. $2\pi \sqrt{\frac{M}{k}}$
• B. $2\pi \sqrt{\frac{M}{k}}$
• C. $2\pi \sqrt{\frac{m+M}{2k}}$
• D. $2\pi \sqrt{\frac{Mm}{k(m+M)}}$

Answer 1

The correct option is D $2\pi \sqrt{\frac{Mm}{k(m+M)}}$

Since the block and wedge will oscillate together, we can use the concept of reduced mass.
Reduced mass $\left(\mu \right)$ of system is given by
$\mu =\frac{mM}{m+M}$
and Time period $\left(T\right)=2\pi \sqrt{\frac{\mu }{k_{eq}}}$
Here, $k_{eq}=k\&\mu =\frac{mM}{m+M}$
So, $T=2\pi \sqrt{\frac{mM}{k(m+M)}}$