Question 3
Two identical solid spheres roll down two different inclined planes of the same height but different angles of inclination. Which sphere reaches the bottom with greater speed?
AI Step-by-Step Solution
Step 1: Energy Conservation
By the law of conservation of mechanical energy, the potential energy at the top is converted into translational and rotational kinetic energy at the bottom.
Step 2: Moment of Inertia
For a solid sphere, the moment of inertia I = ⅖mr². Substitute this and ω = v/r into the energy equation.
mgh = ½mv² + ⅕mv²
gh = 7/10 v²
Step 3: Final Velocity
Solving for v, we get:
The final speed depends only on height h and acceleration due to gravity g. It is independent of the angle of inclination. Therefore, both spheres reach the bottom with the same speed.