A ball rolls without slipping and has radius R and radius of gyration K about its centre. Find the fraction of total kinetic energy that is rotational.
Options

Question

A ball rolls without slipping and has radius R and radius of gyration K about its centre. Find the fraction of total kinetic energy that is rotational.

G1 Physics · Accepted Answer

Step 1: Write the rotational inertia as I = MK² and use pure-rolling condition ω = v/R. Step 2: Rotational kinetic energy is ½Iω² = ½Mv²(K²/R²). Step 3: Total kinetic energy is ½Mv² + ½Mv²(K²/R²) = ½Mv²(1 + K²/R²). Step 4: Fraction rotational = (K²/R²)/(1 + K²/R²) = K²/(K²+R²). Final answer: C — K²/(K²+R²).