MHT CET · Physics · Rotational Motion

A solid sphere of mass $2 kg$ is rolling on a frictionless horizontal surface with velocity $6 m s^{- 1}$ . It collides on the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be (Force constant of the spring = $36 N m^{- 1}$ ).

A $\sqrt{14} m$
B $\sqrt{2.8} m$
C $\sqrt{1.4} m$
D $\sqrt{0.7} m$

Verified Solution

Answer & Solution

Correct Answer

(B) $\sqrt{2.8} m$

Step-by-step Solution

Detailed explanation

Kinetic energy of rolling solid sphere

= \frac{1}{2} m V^{2} + \frac{1}{2} I ω^{2}

= \frac{1}{2} m V^{2} + \frac{1}{2} \times \frac{2}{5} m r^{2} ω^{2}

= \frac{1}{2} m V^{2} + \frac{1}{5} m V^{2}

= \frac{7}{10} m V^{2}

The potential energy of the spring on maximum compression

x

= \frac{1}{2} k x^{2}

∴ \frac{1}{2} k x^{2} = \frac{7}{10} m V^{2}

x^{2} = \frac{14}{10} \frac{m V^{2}}{k}

= \frac{14}{10} \times \frac{2 {(6)}^{2}}{36}

= 2.8

x = \sqrt{2.8} m

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.

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A solid sphere of mass 2 kg is rolling on a frictionless horizontal surface with velocity 6 m s-1. It collides on the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be (Force constant of the spring = 36 N m-1).

Step-by-step Solution

A solid sphere of mass $2 kg$ is rolling on a frictionless horizontal surface with velocity $6 m s^{- 1}$ . It collides on the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be (Force constant of the spring = $36 N m^{- 1}$ ).