In the given figure, two wheels P and Q are connected by a belt B. The radius of P is three times as that of Q. In case of same rotational kinetic energy, the ratio of rotational inertias ( I₁/I₂ ) will be x : 1. The value of x will be _____________.

Solution

Physics sir · 3 steps

The belt does not slip, so both rims move at the same speed. That fixes the ratio of the angular speeds.

1. Same rim speed. Every point of the belt moves at one speed v, and each rim moves with the belt. So v = ω_P (3R) = ω_Q R, which gives ω_Q = 3ω_P. The small wheel spins three times as fast.
2. Equal kinetic energy. Set the two rotational kinetic energies equal: 1/2 I₁ ω_P² = 1/2 I₂ ω_Q². Here I₁ belongs to P and I₂ belongs to Q.

   I₁/I₂ = (ω_Q/ω_P)² = (3ω_P/ω_P)² = 9

3. So. The ratio is 9 : 1, so x = 9.

The common slip: 3. Forgetting to square the angular speed ratio gives 3. Kinetic energy depends on ω², so the factor of 3 becomes 9.

The idea to keep. Wheels joined by a belt share the same rim speed, so ω r is the same for both. With equal 1/2Iω², the moment of inertia goes as 1/ω², which means it goes as r².

Answer from the NTA final answer key. Solution by PrepWiser.