Question

In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. If each spring has spring constant k, the frequency of oscillation of given body is : 

• A. $\frac{1}{2\pi} \sqrt{\frac{k}{Mg \sin \alpha}}$
• B. $\frac{1}{2\pi} \sqrt{\frac{2k}{Mg \sin \alpha}}$
• C. $\frac{1}{2\pi} \sqrt{\frac{2k}{M}}$
• D. $\frac{1}{2\pi} \sqrt{\frac{k}{2M}}$

Hint:

In a spring-mass system, the inclination of the plane affects the equilibrium position (compression/extension) but does not affect the frequency or period of oscillation.

Answer 1

The Correct Option is C

Step 1: The mass is connected to two springs. If the mass is displaced, both springs exert a restoring force in the same direction.
Step 2: This is a parallel combination of springs. The effective spring constant $k_{eff} = k + k = 2k$.
Step 3: The time period is $T = 2\pi \sqrt{\frac{M}{k_{eff}}} = 2\pi \sqrt{\frac{M}{2k}}$.
Step 4: Frequency $f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{2k}{M}}$.