Question

A simple harmonic oscillator having four identical springs (as shown in figure) has a time period \(T_{1}\). Another similar simple harmonic oscillator of the same mass \(m\) but the four identical springs with spring constants \(2k\) has a time period \(T_{2}\), then \(\frac{T_{1}}{T_{2}}\) is

• A. \(2:1\)
• B. \(1:\sqrt{2}\)
• C. \(\sqrt{2}:1\)
• D. \(1:2\)

Hint:

The effective spring constant is proportional to the spring constant of each spring, and the period varies as the inverse square root of it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
\(T=2\pi\sqrt{\dfrac{m}{k_{eff}}}\).
For the same arrangement of four identical springs, \(k_{eff}\propto k\).

Step 2: New spring constant
Replacing k with \(2k\) doubles the effective spring constant: \(k_{eff,2}=2k_{eff,1}\).

Step 3: Ratio
\(\dfrac{T_1}{T_2}=\sqrt{\dfrac{k_{eff,2}}{k_{eff,1}}}=\sqrt{2}\)
So \(T_1:T_2=\sqrt{2}:1\).

Final Answer:
The ratio of time periods is root two to one. \[ \boxed{\text{(C) }\sqrt{2}:1} \]