When a sphere of moment of inertia I about it's centre of gravity, and mass m, rolls from rest down an inclined plane without slipping, it's kinetic energy is calculated from
The kinetic energy of a rolling sphere can be calculated using the following equation:
K = (1/2) Iω² + (1/2) mv²
Where:
- K is the total kinetic energy
- I is the moment of inertia about the center of gravity
- ω is the angular velocity of the sphere (related to its linear velocity v by ω = v/R, where R is the radius of the sphere)
- m is the mass of the sphere
- v is the linear velocity of the sphere
When a sphere rolls without slipping, the linear velocity is related to the angular velocity by v = ωR. Substituting this relationship into the equation, we get:
K = (1/2) Iω² + (1/2) m(ωR)²
Simplifying further:
K = (1/2) Iω² + (1/2) mω²R²
Factoring out ω²:
K = (1/2) ω²(I + mR²)
So the kinetic energy of the rolling sphere is given by (1/2) ω²(I + mR²).