🎡 System of Particles and Rotational Motion

1. Centre of Mass

The point where the entire mass of a body or system of particles is supposed to be concentrated. For a two-particle system on the x-axis:

X_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂)

2. Torque (Moment of Force)

Torque is the turning effect of force. It is the cross product of position vector and force.

🔄 Torque (τ)

τ = r × F = r F sinθ

Unit is Newton-meter (N m). It plays the same role in rotational motion that force plays in linear motion.

3. Angular Momentum (L)

The moment of linear momentum. L = r × p = r p sinθ. Also, L = Iω.

Conservation of Angular Momentum: If net external torque is zero (τ = 0), then L = constant (I₁ω₁ = I₂ω₂). Example: An ice skater spinning faster by pulling her arms in.

4. Moment of Inertia (I)

The rotational analogue of mass. It depends on mass and its distribution from the axis of rotation. I = Σ mr².

Important Theorems

1. Theorem of Perpendicular Axes: Iz = Ix + Iy (Applicable only to planar lamina)

2. Theorem of Parallel Axes: I = I_cm + Md² (Applicable to any body)

5. Rotational Kinematics

Equations are analogous to linear motion:

ω = ω₀ + αt
θ = ω₀t + ½ αt²
ω² = ω₀² + 2αθ