Question

A block of mass $m$ is attached rigidly to a spring of stiffness $k_1$ & touches another spring of stiffness $k_2$ as shown in the figure. Assuming the surface is smooth, the period of oscillation of the block is

• A. $\pi [\sqrt{\frac{m}{k_1}}+\sqrt{\frac{m}{k_1+k_2}}]$
• B. $2\pi \sqrt{\frac{m}{k_1+k_2}}$
• C. $\pi [\sqrt{\frac{m}{k_1}}+\sqrt{\frac{m}{k_2}}]$
• D. $\pi [\sqrt{\frac{m}{k_2}}+\sqrt{\frac{m}{k_1+k_2}}]$

Answer 1

The correct option is A $\pi [\sqrt{\frac{m}{k_1}}+\sqrt{\frac{m}{k_1+k_2}}]$

The block will exceute one half of SHM under the action of spring $k_1$ alone till it touches the spring $k_2$. The other half of SHM will be under the action of both $k_1$ and $k_2$ which are parallel.
$k_{eq}=k_1+k_2$
Time period of first half
$t_1=\frac{T_1}{2}=\frac{1}{2}\left(2\pi \sqrt{\frac{m}{k_1}}\right)=\pi \sqrt{\frac{m}{k_1}}$
& Time period of second half
$t_2=\frac{T_2}{2}=\frac{1}{2}\left(2\pi \sqrt{\frac{m}{k_{eq}}}\right)$
$=\pi \sqrt{\frac{m}{k_1+k_2}}$
$\therefore$ Total time period of SHM $t=t_1+t_2$
$\Rightarrow t=\pi [\sqrt{\frac{m}{k_1}}+\sqrt{\frac{m}{k_1+k_2}}]$
Hence 'A' is the correct option.