Question

A body executes SHM under the influence of one force and has a time period of \(T_1\) seconds. The same body executes SHM with a time period of \(T_2\) seconds under the influence of another force separately. When both forces act simultaneously and in the same direction, then the time period of the body is

• A. \[ (T_1+T_2)\ \text{sec} \]
• B. \[ \sqrt{T_1^2+T_2^2}\ \text{sec} \]
• C. \[ \sqrt{\frac{T_1+T_2}{T_1T_2}}\ \text{sec} \]
• D. \[ \sqrt{\frac{T_1^2T_2^2}{T_1^2+T_2^2}}\ \text{sec} \]

Hint:

When two restoring forces act simultaneously, \[ k_{\text{eff}}=k_1+k_2. \] Using \[ T=2\pi\sqrt{\frac{m}{k}}, \] one gets \[ \frac1{T^2} = \frac1{T_1^2} + \frac1{T_2^2}. \]

Answer 1

The Correct Option is D

Concept: For SHM, \[ T=2\pi\sqrt{\frac{m}{k}}, \] where \(k\) is the effective force constant. When two restoring forces act simultaneously in the same direction, the effective force constant becomes \[ k=k_1+k_2. \]

Step 1: Express \(k_1\) and \(k_2\) in terms of \(T_1\) and \(T_2\). For the first SHM, \[ T_1=2\pi\sqrt{\frac{m}{k_1}}. \] Squaring, \[ T_1^2=\frac{4\pi^2m}{k_1}. \] Hence, \[ k_1=\frac{4\pi^2m}{T_1^2}. \] Similarly, \[ k_2=\frac{4\pi^2m}{T_2^2}. \]

Step 2: Find the effective force constant. \[ k=k_1+k_2. \] \[ k = 4\pi^2m \left( \frac1{T_1^2} + \frac1{T_2^2} \right). \]

Step 3: Find the new time period. \[ T = 2\pi\sqrt{\frac{m}{k}}. \] Substituting \(k\), \[ T = 2\pi \sqrt{ \frac{m} {4\pi^2m\left(\frac1{T_1^2}+\frac1{T_2^2}\right)} }. \] \[ T = \frac{1} {\sqrt{\frac1{T_1^2}+\frac1{T_2^2}}}. \] \[ T = \frac{1} {\sqrt{\frac{T_1^2+T_2^2}{T_1^2T_2^2}}}. \] \[ T = \sqrt{ \frac{T_1^2T_2^2} {T_1^2+T_2^2} }. \] Therefore, \[ \boxed{ T= \sqrt{ \frac{T_1^2T_2^2} {T_1^2+T_2^2} } } \] \[ \boxed{\text{Answer = (D)}} \]