Question

On a smooth inclined plane, a block of mass $M$ is fixed to two rigid supports using two springs, each having spring constant $k$, as shown in the figure. If the masses of the springs are neglected, then the period of oscillation of the block is

• A. \(2\pi \sqrt{\frac{M}{2k}}\)
• B. \(2\pi \sqrt{\frac{2M}{k}}\)
• C. \(2\pi \sqrt{\frac{Mg \sin \theta}{2k}}\)
• D. \(2\pi \sqrt{\frac{2Mg}{k}}\)

Hint:

When two springs are attached in parallel, their spring constants add. Use the effective spring constant to find the oscillation period.

Answer 1

The Correct Option is A

Each spring has spring constant \(k\), and they are connected in parallel to the block. Effective spring constant is: \[ k_{\text{eff}} = k + k = 2k \] The formula for period of oscillation for mass-spring system is: \[ T = 2\pi \sqrt{\frac{M}{k_{\text{eff}}}} = 2\pi \sqrt{\frac{M}{2k}} \] Thus, the period of oscillation of the block is \(2\pi \sqrt{\frac{M}{2k}}\).