A sphere rolls down an inclined plane without slipping. What fraction of its total energy is rotational ?

Calculation:
 A sphere rolling without slipping has both translational and rotational kinetic energy.

Rotational Kinetic Energy (K_R) = (1/2) × I × (v² / r²)

Moment of Inertia of a solid sphere, I = (2/5)mr²

So, K_R = (1/2) × (2/5)mr² × (v² / r²) = (1/5)mv²

Translational Kinetic Energy (K_Tr) = (1/2)mv²

Total Energy (K_Total) = K_R + K_Tr = (1/5)mv² + (1/2)mv²

Convert to common denominator:

K_Total = (2/10 + 5/10)mv² = (7/10)mv²

Fraction of Rotational Energy = K_R / K_Total = (1/5)mv² / (7/10)mv²

Cancel mv²:

Fraction = (1/5) / (7/10) = (2/7)

Answer: Option 1) 2/7