JEE Advanced 2022 · Paper 2 · Physics Q7

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PhysicsInteger+3 / -1Question 25 of 54

On a frictionless horizontal plane, a bob of mass $m = 0.1$ kg is attached to a spring with natural length $l_0 = 0.1$ m. The spring constant is $k_1 = 0.009\ \mathrm{N\,m^{-1}}$ when the length of the spring $l > l_0$ and is $k_2 = 0.016\ \mathrm{N\,m^{-1}}$ when $l < l_0$ . Initially the bob is released from $l = 0.15$ m. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T = (n\pi)$ s, then the integer closest to $n$ is _____.

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Answer and worked solution

Official answer: 6

The motion is two half-oscillations: one with the spring stretched (constant $k_1$ ), one with it compressed (constant $k_2$ ). Each half takes half the period of its own SHM.

Half-periods.

Stretched: $\pi\sqrt{\dfrac m{k_1}}=\pi\sqrt{\dfrac{0.1}{0.009}}=\dfrac{10}3\pi$ s.
Compressed: $\pi\sqrt{\dfrac m{k_2}}=\pi\sqrt{\dfrac{0.1}{0.016}}=2.5\pi$ s.

Total. $T=\left(\frac{10}3+2.5\right)\pi\approx5.83\pi$ s, so the integer closest to $n$ is 6.

Topics:Spring-mass systems Simple harmonic motion

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