GATE 2023 ME – Question 30

Vibrations · Free vibration of single degree of freedom systems, damping · 1 mark · Numerical answer

The figure shows a block of mass m = 20 kg attached to a pair of identical linear springs, each having a spring constant k = 1000 N/m. The block oscillates on a frictionless horizontal surface. Assuming free vibrations, the time taken by the block to complete ten oscillations is _________ seconds. (Rounded off to two decimal places)
Take $\pi = 3.14$.

a block of mass m on a horizontal surface, connected to a wall by two identical springs side by side, one above the other.

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Correct answer: 6.25 to 6.31

Explanation

Both springs are attached between the wall and the block, so they act in parallel and the equivalent stiffness is $k_{eq} = 2k = 2000$ N/m. The natural frequency is $\omega_n = \sqrt{\frac{2000}{20}} = 10$ rad/s. The period is $T = \frac{2\pi}{\omega_n} = \frac{2 \times 3.14}{10} = 0.628$ s, so ten oscillations take $6.28$ s.