Learning Objectives
- Locate the centre of mass of a system of particles and rigid bodies
- Understand torque and angular momentum
- Learn about moment of inertia and its theorems
- Apply the equations of rotational kinematics and dynamics
- Understand rolling motion as combined translation and rotation
Key Concepts
Centre of Mass
The point where the entire mass of a system can be considered to be concentrated.
For discrete particles: X_cm = Σmᵢxᵢ / Σmᵢ, similarly for Y_cm and Z_cm.
For continuous body: X_cm = (1/M)∫x dm
The centre of mass moves as if all external forces act on it: F_ext = Ma_cm
Common centres of mass: uniform rod (L/2), circular disc (centre), triangle (centroid at h/3), semicircular disc (4R/3π from centre).
Torque (Moment of Force)
τ = r × F (cross product of position vector and force).
Magnitude: τ = rF sin θ. SI unit: N·m. It is a vector (direction by right-hand rule).
Torque about an axis: τ = Iα (analogous to F = ma)
Angular Momentum
L = r × p = r × mv
For a rigid body rotating about a fixed axis: L = Iω
Relation: τ = dL/dt (Newton's second law for rotation)
Conservation of Angular Momentum: If net external torque is zero, L = Iω = constant. This explains why an ice skater spins faster when arms are pulled in.
Moment of Inertia
Rotational analogue of mass: I = Σmᵢrᵢ² (for discrete particles) or I = ∫r² dm (continuous body).
SI unit: kg·m². It depends on the axis of rotation, mass, and distribution of mass.
Moments of inertia of common bodies (about standard axes):
- Thin rod (about centre): ML²/12; (about end): ML²/3
- Solid sphere (about diameter): (2/5)MR²
- Hollow sphere: (2/3)MR²
- Solid cylinder (about axis): MR²/2
- Hollow cylinder: MR²
- Disc (about axis): MR²/2; (about diameter): MR²/4
- Ring (about axis): MR²; (about diameter): MR²/2
Theorems of Moment of Inertia
Parallel Axis Theorem: I = I_cm + Md² (I about any axis = I about parallel axis through CM + Md²).
Perpendicular Axis Theorem: I_z = I_x + I_y (only for planar bodies; z-axis perpendicular to plane).
Rotational Kinematics
ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ (analogous to linear equations).
Rotational Kinetic Energy
KE_rot = ½Iω²
Rolling Motion: KE_total = ½mv² + ½Iω² = ½mv²(1 + k²/R²), where k is radius of gyration.
For rolling without slipping: v = Rω
Velocity at bottom of incline: v = √(2gh/(1 + k²/R²))
Summary
The centre of mass is a key concept for analyzing system motion. Torque is the rotational analogue of force, and angular momentum is the rotational analogue of linear momentum. Moment of inertia quantifies rotational inertia and depends on mass distribution. The parallel and perpendicular axis theorems help calculate moments about different axes. Rolling motion combines translation and rotation.
Important Terms
- Centre of Mass: Point representing average position of mass distribution
- Torque: Rotational effect of a force, τ = r × F
- Moment of Inertia: Rotational analogue of mass, I = Σmᵢrᵢ²
- Angular Momentum: L = Iω, conserved when net torque = 0
- Radius of Gyration: k = √(I/M), distance at which entire mass can be concentrated to give same I
Quick Revision
- X_cm = Σmᵢxᵢ/M; F_ext = Ma_cm
- τ = r × F = Iα; L = Iω; τ = dL/dt
- Parallel axis: I = I_cm + Md²; Perpendicular axis: I_z = I_x + I_y
- KE_rolling = ½mv²(1 + k²/R²)
- Conservation: if τ_ext = 0, then Iω = constant
- Solid sphere rolls fastest down an incline (smallest k²/R² = 2/5)