MHT CET · Physics · Oscillations

All the springs in fig. (a), (b) and (c) are identical, each having force constant K. Mass attached to each system is ' \(m\) '. If \(T_a, T_b\) and \(T_c\) are the time periods of oscillations of the three systems respectively, then

A \(\mathrm{T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{b}}\)
B \(T_a=\frac{T_c}{\sqrt{2}}\)
C \(\mathrm{T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{a}}\)
D \(T_b=2 T_c\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(T_b=2 T_c\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned}
T_a & =2 \pi \sqrt{\frac{m}{K}} \\
T_b & =2 \pi \sqrt{\frac{m}{\frac{K}{2}}} \\
\therefore \quad T_b & =2 \pi \sqrt{\frac{2 m}{K}}=\sqrt{2} T_a \Rightarrow T_a=\frac{T_b}{\sqrt{2}}...(i) \\
T_c & =2 \pi \sqrt{\frac{m}{2 K}}=\frac{T_a}{\sqrt{2}} \Rightarrow T_a=\sqrt{2} T_c
...(ii)\end{aligned}\)
From (i) and (ii),
\(\frac{\mathrm{T}_{\mathrm{b}}}{\sqrt{2}}=\sqrt{2} \mathrm{~T}_{\mathrm{c}} \Rightarrow \mathrm{~T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{c}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app