Rolling Motion

From NCERT Class 11, Chapter 7 (System of Particles and Rotational Motion), a body rolling without slipping down an incline has linear acceleration a = g sinθ / (1 + k²/R²), where k is the radius of gyration of the body about its central axis. The denominator captures the fraction of gravitational potential energy that must be diverted into rotational kinetic energy rather than translation. A smaller value of k²/R² therefore means more energy goes into forward motion and the body accelerates faster down the slope. For a solid sphere the shape factor is k²/R² = 2/5, while for a hollow thin-walled sphere it is k²/R² = 2/3, because the hollow body keeps all its mass at the outer radius. Since 2/5 is less than 2/3, the solid sphere has the larger acceleration and reaches the bottom of the incline first. The option 'The hollow sphere' is wrong because its larger rotational inertia slows its descent. The option 'Both reach together' ignores the difference in mass distribution, which is decisive here. The option 'Depends on the mass' is incorrect because the mass cancels out of the acceleration expression entirely. A check confirms that the result depends only on the dimensionless shape factor k²/R², not on the mass or the radius, which is exactly why two bodies of identical mass and radius can still finish at different times.