As shown in the figure, $S_1$ and $S_2$ are identical springs with spring constant K each. The oscillation frequency of the mass 'm' is 'f'. If the spring $S_2$ is removed, the oscillation frequency will become ______.
Show Hint
It is highly counter-intuitive, but when a mass is trapped in the middle of two springs anchored to opposite walls, those springs are in parallel, not series! They both push/pull the mass back toward the center simultaneously, doubling the effective stiffness ($2K$).
Step 1: Understanding the Question:
We have a block of mass $m$ oscillating while attached to two identical springs. (Based on standard physics problems of this type, when a mass is flanked by two springs attached to rigid walls, or hung from two springs side-by-side, the springs are functionally in parallel). We need to determine how removing one spring changes the oscillation frequency. Step 2: Key Formula or Approach:
1. The frequency of a spring-mass system is $f = \frac{1}{2\pi} \sqrt{\frac{K_{eq}}{m}}$.
2. For two springs in parallel (which is standard when a mass is between two walls, as displacing the mass by $x$ compresses one spring by $x$ and stretches the other by $x$, creating a restoring force from BOTH), the equivalent stiffness is $K_{eq} = K_1 + K_2$. Step 3: Detailed Explanation:
Initial Case (Both springs present):
Since $S_1$ and $S_2$ are identical ($K_1 = K_2 = K$) and act in parallel:
$$K_{initial} = K + K = 2K$$
The initial frequency is given as $f$:
$$f = \frac{1}{2\pi} \sqrt{\frac{2K}{m}}$$
Final Case (Spring $S_2$ removed):
Now the mass is attached to only one spring ($S_1$).
$$K_{final} = K$$
The new frequency $f'$ is:
$$f' = \frac{1}{2\pi} \sqrt{\frac{K}{m}}$$
To find the relationship, let's look at $f$ again:
$$f = \frac{1}{2\pi} \sqrt{\frac{2K}{m}} = \sqrt{2} \left( \frac{1}{2\pi} \sqrt{\frac{K}{m}} \right)$$
Notice the term in the parenthesis is exactly $f'$:
$$f = \sqrt{2} \cdot f'$$
Isolate $f'$:
$$f' = \frac{f}{\sqrt{2}}$$ Step 4: Final Answer:
The new frequency becomes $f/\sqrt{2}$, matching option (c).
