Question

Two ideal springs of different lengths have been suspended from points $A$ and $B$ such that their free ends $C$ and $D$ are at the same horizontal level. A massless rod $PQ$ is attached to the ends of the springs. A block of mass $m$ is attached to the rod at point $R$. The rod remains horizontal in equilibrium. Now, the block is pulled down and released. It performs vertical oscillations with time period $T=2\pi \sqrt{\frac{m}{3k}}$ where $k$ is the force constant of the longer spring.


Find the ratio of lengths $RC$ and $RD$.

• A. $3:1$
• B. $3:2$
• C. $2:1$
• D. $4:3$

Answer 1

The correct option is C $2:1$

From the data given in the question, time period of vertical oscillations is $T=2\pi \sqrt{\frac{m}{3k}}$
From this, effective force constant is $k_{eq}=3k$.
Since the springs $BD$ and $AC$ are parallel,
$k_{eq}=k_{AC}+k_{BD}=3k$
Spring $BD$ must have a force constant of $2k$.
Given that, rod $PQ$ is massless. So, net force and net torque on it must be zero.


Net force $F=0\Rightarrow mg=3kx$
Now, Torque about $R$
$kx\times RC-2kx\times RD=0$
$\Rightarrow \frac{RC}{RD}=\frac{2}{1}$
Thus, option (c) is the correct answer.