Question

Two springs of force constants '\(2k\)' and '\(k\)' are connected to a mass '\(m\)' as shown. Mass is displaced slightly to one side and released. The frequency of oscillation of the two springs-mass system is

• A. \(\frac{1}{2π}\sqrt{\frac{m}{k}}\)
• B. \(\frac{1}{2π}\sqrt{\frac{k}{m}}\)
• C. \(\frac{1}{2π}\sqrt{\frac{2k}{m}}\)
• D. \(\frac{1}{2π}\sqrt{\frac{3k}{m}}\)

Hint:

Springs on both sides of a mass act in parallel, so their constants add.

Answer 1

The Correct Option is D

Step 1: Understanding the Figure:
The mass is attached to a spring of constant \(2k\) on the left wall and a spring of constant \(k\) on the right wall. When the mass moves, one spring stretches while the other compresses, and both give a restoring force in the same direction.

Step 2: Effective constant:
The two springs act in parallel, so \(k_{\text{eff}} = 2k + k = 3k\).

Step 3: Frequency:
\[ f = \frac{1}{2\pi}\sqrt{\frac{k_{\text{eff}}}{m}} = \frac{1}{2\pi}\sqrt{\frac{3k}{m}} \]

Step 4: Why the other options are wrong.
\(\frac{1}{2\pi}\sqrt{\frac mk}\) has \(m\) and \(k\) inverted. Options \(\sqrt{\frac km}\) and \(\sqrt{\frac{2k}{m}}\) use only one spring or a wrong sum.

Final Answer:
The frequency is \(\frac{1}{2\pi}\sqrt{\frac{3k}{m}}\), option (D). \[ \boxed{\frac{1}{2\pi}\sqrt{\frac{3k}{m}}} \]