AIIMS 2003 Physics Spring Mass System MCQ Question

To determine the frequency of oscillation of a mass connected to two springs with force constants $k$ and $2k$, we first need to find the equivalent spring constant when the springs are arranged in parallel.

Step 1: Understanding the Configuration

Assuming the two springs are connected in parallel to the mass, the equivalent spring constant $K$ can be calculated as:

$$K = k_1 + k_2$$

where $k_1 = k$ (the spring constant of the first spring) and $k_2 = 2k$ (the spring constant of the second spring). Therefore, we have:

$$K = k + 2k = 3k$$

Step 2: Frequency of Oscillation

The frequency of oscillation $f$ of a mass-spring system is given by the formula:

$$f = \frac{1}{2\pi} \sqrt{\frac{K}{m}}$$

where $K$ is the equivalent spring constant and $m$ is the mass attached to the springs. Substituting the equivalent spring constant we found:

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

Conclusion

Thus, the frequency of oscillation of the mass is:

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

This matches with option C, confirming that the correct answer is C) $\frac{1}{2\pi} \sqrt{\frac{3k}{m}}$.

Explanation of Incorrect Options

Now, let's clarify why the other options are incorrect:

• Option A: $\frac{1}{2\pi} \sqrt{\frac{k}{m}}$

  This option would be valid if there were only one spring with a spring constant $k$. However, since we have two springs with constants $k$ and $2k$, this does not account for the combined effect of both springs.

• Option B: $\frac{1}{2\pi} \sqrt{\frac{2k}{m}}$

  This would be true if the equivalent spring constant was $2k$, which again neglects the contribution of the spring $k$. Thus, it only considers one spring.

• Option D: $\frac{1}{2\pi} \sqrt{\frac{k}{m}}$

  Similar to option A, this option only considers the spring with spring constant $k$ and ignores the second spring, leading to an incorrect calculation.

Summary

To summarize, the calculated equivalent spring constant for the two springs in parallel is $3k$. Therefore, the frequency of oscillation of the mass is given by:

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

This confirms that the correct answer is option C) $\frac{1}{2\pi} \sqrt{\frac{3k}{m}}$.