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AI Review

Rotational Dynamics DPP #4

Score: 14/20

Question 3

Medium Difficulty

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?

C
They reach with the same speed. Your Answer
{/* AI Solution Breakdown */}

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.

mgh = ½mv² + ½Iω²

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 + ½(⅖mr²)(v²/r²)
mgh = ½mv² + ⅕mv²
gh = 7/10 v²

Step 3: Final Velocity

Solving for v, we get:

v = √(10gh/7)

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.

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