JEE Main:Rotational Motion: Set-8

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🎯 Rotational Kinematics Problems

1. A wheel starts from rest and accelerates uniformly at 2 rad/s² for 10 seconds. Calculate the angular displacement during this time.

2. A flywheel rotating at 300 rpm is brought to rest by a constant angular deceleration in 15 seconds. Find:
(a) The angular deceleration
(b) The number of revolutions made before coming to rest

3. A disc of radius 0.5 m and moment of inertia 2 kg⋅m² is rotating at 120 rpm. A tangential force of 10 N is applied at its rim for 5 seconds. Calculate the final angular velocity.

⚙️ Advanced Problems

4. A solid cylinder of mass 5 kg and radius 0.2 m rolls down an inclined plane of height 3 m without slipping. Find the linear velocity at the bottom of the incline. (g = 10 m/s²)

5. Two discs A and B have moments of inertia I₁ = 3 kg⋅m² and I₂ = 2 kg⋅m² respectively. Disc A rotates at 100 rad/s while disc B is at rest. They are brought into contact so that they rotate together. Calculate:
(a) The common angular velocity
(b) The loss in kinetic energy

📋 Answer Key

1. θ = ½αt² = ½ × 2 × 10² = 100 rad

2. ω₀ = 300 rpm = 10π rad/s
(a) α = (ω - ω₀)/t = (0 - 10π)/15 = -2π/3 rad/s²
(b) θ = ω₀t + ½αt² = 10π × 15 + ½ × (-2π/3) × 15² = 75π rad = 37.5 revolutions

3. τ = Fr = 10 × 0.5 = 5 N⋅m
α = τ/I = 5/2 = 2.5 rad/s²
ω₀ = 120 rpm = 4π rad/s
ωf = ω₀ + αt = 4π + 2.5 × 5 = 4π + 12.5 ≈ 25.07 rad/s

4. For rolling motion: v² = (2gh)/(1 + I/mR²)
For solid cylinder: I = ½mR²
v² = (2 × 10 × 3)/(1 + ½) = 60/1.5 = 40
v = 2√10 ≈ 6.32 m/s

5. (a) Conservation of angular momentum: I₁ω₁ = (I₁ + I₂)ωf
3 × 100 = (3 + 2)ωf → ωf = 60 rad/s
(b) KEᵢ = ½I₁ω₁² = ½ × 3 × 100² = 15,000 J
KEf = ½(I₁ + I₂)ωf² = ½ × 5 × 60² = 9,000 J
Loss = 15,000 - 9,000 = 6,000 J