Question

Two identical springs of constant $K$ are connected in series and parallel in shown in figure. A mass $M$ is suspended from them. The ratio of their frequencies is series to parallel combination will be

• A. 1 : 2
• B. 1 : 4
• C. 4 : 1
• D. 1 : $\sqrt{2}$

Hint:

Spring constants behave exactly opposite to electrical resistors. Series arrangements weaken the effective constant ($K/2$), while parallel arrangements strengthen it ($2K$). Since frequency depends on $\sqrt{k_{eq}}$, the parallel setup will always be $\sqrt{4} = 2$ times faster than the series setup.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the ratio of oscillation frequencies when a mass $M$ is attached to two identical springs of spring constant $K$ arranged in two configurations: series and parallel.
Note: While the exact image code placeholder is maintained, the structural configurations of standard series and parallel spring setups follow well-established physics rules.

Step 2: Key Formula or Approach:
The frequency of a spring-mass system undergoing simple harmonic motion is given by:
$$f = \frac{1}{2\pi}\sqrt{\frac{k_{eq}}{M}}$$ Thus, the frequency is directly proportional to the square root of the equivalent spring constant: $f \propto \sqrt{k_{eq}}$.
For two identical springs with constant $K$:
1. In a series combination, the equivalent spring constant is: $$\frac{1}{k_s} = \frac{1}{K} + \frac{1}{K} = \frac{2}{K} \implies k_s = \frac{K}{2}$$ 2. In a parallel combination, the equivalent spring constant is: $$k_p = K + K = 2K$$

Step 3: Detailed Explanation:
Let's express the frequency of oscillation for both cases using our formulas.
For the series combination, substituting $k_s = \frac{K}{2}$:
$$f_1 = \frac{1}{2\pi}\sqrt{\frac{K}{2M}}$$ For the parallel combination, substituting $k_p = 2K$:
$$f_2 = \frac{1}{2\pi}\sqrt{\frac{2K}{M}}$$ Now, take the ratio of the series configuration frequency ($f_1$) to the parallel configuration frequency ($f_2$):
$$\frac{f_1}{f_2} = \frac{\frac{1}{2\pi}\sqrt{\frac{K}{2M}}}{\frac{1}{2\pi}\sqrt{\frac{2K}{M}}}$$ The common constant terms $\frac{1}{2\pi}$, $K$, and $M$ cancel out neatly:
$$\frac{f_1}{f_2} = \sqrt{\frac{1/2}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$$ Therefore, the ratio of their frequencies is 1 : 2.

Step 4: Final Answer:
The ratio of frequencies from series to parallel combination is 1 : 2, corresponding to option (A).