Question

In the figure all springs are identical having spring constant k and mass m each. The block also has mass m. The frequency of oscillation of the block is:

• A. $\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$
• B. $\frac{1}{2\pi}\sqrt{\frac{3k}{2m}}$
• C. $2\pi\sqrt{\frac{3m}{3k}}$
• D. none of these

Answer 1

Answer: (B)

$\frac{1}{2\pi}\sqrt{\frac{3k}{2m}}$

Answer 2

The frequency of oscillation can be derived as follows:

1. Equivalent spring constant:

   • The two top springs in parallel have an effective spring constant: $k_{\text{top}} = k + k = 2k$.

   • The block is acted on by the top set and the bottom spring, so the net effective spring constant is: $k_{\text{eff}} = 2k + k = 3k$.

2. Effective mass:

   • For each spring of mass $m$, the effective inertial contribution is $\frac{1}{3}m$ (for a uniformly distributed mass in a spring with one end fixed).

   • Total effective mass contributed by the three springs: $m_{\text{springs}} = 3 \times \frac{m}{3} = m$.

   • Adding the block's mass, the total effective mass is: $M = m + m = 2m$.

3. Frequency of oscillation:

   The angular frequency is given by $\omega = \sqrt{\frac{k_{\text{eff}}}{M}} = \sqrt{\frac{3k}{2m}}$. Thus, the frequency is $f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{3k}{2m}}$.