Question:
The springs are connected to the blocks as shown in figures A and B. When the blocks are slightly displaced and released, they oscillate with time periods T_A and T_B respectively. Then, the value of T_AT_B is:
The springs are connected to the blocks as shown in figures A and B. When the blocks are slightly displaced and released, they oscillate with time periods T_A and T_B respectively. Then, the value of T_AT_B is:
Show Hint
Whenever two springs act on a block from opposite sides, both forces act in the same restoring direction. So always treat them as parallel:
\[
K_{\text{eq}} = K_1 + K_2
\]
Updated On: Jun 10, 2026
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The Correct Option is A
Solution and Explanation
Concept:
For a mass-spring system undergoing SHM, the time period is:
\[
T = 2\pi \sqrt{\frac{m}{K_{\text{eq}}}}
\]
Step 1: System A analysis In system A: - Mass = 3m - Two identical springs of constant K act on both sides. When displaced by x: Left spring force = Kx (restoring) Right spring force = Kx (restoring) Total restoring force: \[ F = -(Kx + Kx) = -2Kx \] So, \[ K_A = 2K \] Time period: \[ T_A = 2\pi \sqrt{\frac{3m}{2K}} \]
Step 2: System B analysis In system B: - Mass = m - Springs = K and 2K Total restoring force: \[ F = -(K + 2K)x = -3Kx \] So, \[ K_B = 3K \] Time period: \[ T_B = 2\pi \sqrt{\frac{m}{3K}} \]
Step 3: Ratio \[ \frac{T_A}{T_B} = \sqrt{\frac{3m}{2K} \cdot \frac{3K}{m}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \]
Step 1: System A analysis In system A: - Mass = 3m - Two identical springs of constant K act on both sides. When displaced by x: Left spring force = Kx (restoring) Right spring force = Kx (restoring) Total restoring force: \[ F = -(Kx + Kx) = -2Kx \] So, \[ K_A = 2K \] Time period: \[ T_A = 2\pi \sqrt{\frac{3m}{2K}} \]
Step 2: System B analysis In system B: - Mass = m - Springs = K and 2K Total restoring force: \[ F = -(K + 2K)x = -3Kx \] So, \[ K_B = 3K \] Time period: \[ T_B = 2\pi \sqrt{\frac{m}{3K}} \]
Step 3: Ratio \[ \frac{T_A}{T_B} = \sqrt{\frac{3m}{2K} \cdot \frac{3K}{m}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \]
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