When a ceiling fan is switched off, its angular velocity falls to ( 1 3 )^ rd while it makes 24 rotations. How many more rotations will it make before coming to rest?

Step-by-step solution

Define angular velocity and deceleration: Let ₀ be the initial angular velocity and the constant angular deceleration. Phase 1: Velocity reduction to ₀ 3 over 24 rotations. Using the kinematic relation ^2 = ₀^2 + 2 , where = ₀ 3 and = 24 rotations: ( ₀ 3 )^2 = ₀^2 + 2 (24) ₀^2 9 - ₀^2 = 48 - 8 9 ₀^2 = 48 = - ₀^2 54 rotations ⁻¹ Phase 2: Rest from ₀ 3 . With ' = ₀ 3 , '' = 0 , and as above, solve for additional rotations ₂ : 0 = ( ₀ 3 )^2 + 2 (- ₀^2 54 ) ₂ ₀^2 9 = ₀^2 27 ₂ ₂ = 3 rotations The fan makes 3 additional