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NEET PHYSICSMedium

A hollow sphere of diameter 0.2 m0.2 \text{ m} and mass 2 kg2 \text{ kg} is rolling on an inclined plane with velocity v=0.5 m/sv = 0.5 \text{ m/s}. The kinetic energy of the sphere is:

A

0.1 J0.1 \text{ J}

B

0.3 J0.3 \text{ J}

C

0.5 J0.5 \text{ J}

D

0.42 J0.42 \text{ J}

Step-by-Step Solution

The total kinetic energy of a rolling body is the sum of its translational and rotational kinetic energies. Ktotal=Ktrans+Krot=12mv2+12Iω2K_{total} = K_{trans} + K_{rot} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 For a hollow sphere, the moment of inertia about an axis passing through its centre is I=23mR2I = \frac{2}{3}mR^2. For pure rolling, ω=vR\omega = \frac{v}{R}. Therefore, Ktotal=12mv2+12(23mR2)(vR)2=12mv2+13mv2=56mv2K_{total} = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{2}{3}mR^2\right)\left(\frac{v}{R}\right)^2 = \frac{1}{2}mv^2 + \frac{1}{3}mv^2 = \frac{5}{6}mv^2 Given m=2 kgm = 2 \text{ kg} and v=0.5 m/sv = 0.5 \text{ m/s}. Ktotal=56×2×(0.5)2=56×2×0.25=2.560.4167 J0.42 JK_{total} = \frac{5}{6} \times 2 \times (0.5)^2 = \frac{5}{6} \times 2 \times 0.25 = \frac{2.5}{6} \approx 0.4167 \text{ J} \approx 0.42 \text{ J}.

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