JEE Level: 1. Particle in a vertical circular motion: Formula: Tension at top, T_t = mg Tension at bottom, T_b = mg + mv^2/r (a) v = 1 m/s T_t = 1 kg × 9.8 m/s² = 9.8 N T_b = 1 kg × 9.8 m/s² + (1 kg × (1 m/s)² / 1 m) = 11.8 N

(b) v = 2 m/s T_t = 1 kg × 9.8 m/s² = 9.8 N T_b = 1 kg × 9.8 m/s² + (1 kg × (2 m/s)² / 1 m) = 13.8 N

2. Rotational dynamics of a uniform rod: Formula: Angular acceleration, α = F/I Moment of inertia for a rod about one end, I = (1/3)ML^2 F = 10 N M = 2 kg L = 1 m I = (1/3) × 2 kg × (1 m)² = 2/3 kg m² α = 10 N / (2/3 kg m²) = 15 rad/s²

3. Angular momentum of a rotating disk: Formula: Angular momentum, L = Iω Moment of inertia for a disk, I = (1/2)MR² M = 10 kg R = 2 m ω = 1000 rpm = 1000 × 2π/60 rad/s = 104.72 rad/s I = (1/2) × 10 kg × (2 m)² = 20 kg m² L = 20 kg m² × 104.72 rad/s = 2094.4 kg m²/s

CBSE Board Level:

1. Torque required to stop a rotating disc: Formula: Torque, τ = Iα Angular acceleration, α = -ω/T Moment of inertia for a disk, I = (1/2)MR² M = mass of the disc R = radius of the disc ω = angular velocity of the disc T = time taken to stop the disc τ = (1/2)MR² × (-ω/T) = -(1/2)MRω/T

2. Angular frequency of a particle-spring system: Formula: Angular frequency, ω = √(k/m) k = spring constant m = mass of the particle

3. Tension in strings supporting a uniform rod: Formula: Tension in each string, T = (Mg/2)sinθ M = mass of the rod L = length of the rod θ = angle made by each string with the vertical