System of Particles & Rotational Motion

Think of a cricket bat. You hit ball, bat doesn't just go forward, it also spins in your hands. Or a rolling cycle wheel - it moves forward AND spins. Till now we studied point objects - one dot. Real objects have size, they can translate + rotate both.

This chapter is about that. How to handle many particles together? Where is the balancing point? Why door handle is far from hinge? Why ice skater spins faster when she pulls arms in? Why does a ring roll slower than a solid sphere down same slope? All answers here.

Let's start from zero, like you never saw this before.

Take two friends on a see-saw. One 60kg, one 30kg. If both sit at same distance, heavy side goes down. To balance, lighter friend sits farther. The balancing point is centre of mass.

Formula for two particles: x_cm = (m1 x1 + m2 x2) / (m1+m2). Weighted average, heavier pulls COM towards itself.

Example: m1=1kg at x=0, m2=2kg at x=3m. COM = (1*0 + 2*3)/3 = 2m from origin, closer to 2kg. Just like that.

For many particles: r_cm = (m1 r1 + m2 r2 + ...)/M total. For continuous body like rod, R_cm = (1/M) ∫ r dm.

Uniform bodies: Uniform rod COM at middle L/2, uniform ring at centre, uniform disc at centre, triangle at centroid intersection of medians, cone at h/4 from base. Your cricket bat COM is not middle because weight distribution different - heavier bottom.

Most important law: Motion of COM: M a_cm = F_ext total external force. P_total = M V_cm. Means system behaves as if whole mass M is at COM and all external forces act there. Internal forces like you pushing inside boat cancel.

If F_ext=0, V_cm constant, COM moves with constant velocity or stays. Classic: Man in boat. Man walks forward, boat moves backward such that COM of man+boat stays at same place (if no water resistance). That's why you slip when you jump from boat to shore - boat goes back.

Centre of gravity is where weight acts. If g uniform (usual), CG = COM. If g varies (tall building), slightly different but for us same.

In translation, mass resists acceleration - F=ma. More mass harder to accelerate. In rotation, what resists angular acceleration? Moment of Inertia I.

But I depends not just on mass, but WHERE mass is from axis. Same mass farther = more inertia. Like you have dumbbell close to body easy to spin, arms stretched hard to spin. That's why figure skater arms in fast, arms out slow.

Definition: For point masses I = Σ m_i r_i² - mass times distance squared from axis. For continuous ∫ r² dm. Unit kg m².

Radius of gyration K: Imagine whole mass M at distance K from axis gives same I. So I = M K² => K = √(I/M). K tells how far mass is effectively.

Two super important theorems fact-checked:

1) Parallel Axis Theorem: I = I_cm + M d². I about any axis parallel to axis through COM = I_cm + M times distance squared between axes. Example: Rod about centre 1/12 M L², about end distance d=L/2, so I_end = 1/12 M L² + M (L/2)² = 1/12 + 1/4 = 1/3 M L². That's correct.

2) Perpendicular Axis Theorem: For planar lamina (flat 2D like paper), I_z = I_x + I_y where z perpendicular to plane, x,y in plane through same point. So if you know two in-plane, you get perpendicular. Valid only for flat bodies.

Standard bodies table (double-checked, these are correct for uniform bodies):
- Rod length L mass M about centre ⊥ to rod: 1/12 M L². About end ⊥: 1/3 M L².
- Ring radius R mass M about centre ⊥ to plane: M R² (all mass at R). About diameter in plane: ½ M R².
- Disc radius R about centre ⊥: ½ M R². About diameter: ¼ M R².
- Solid sphere about any diameter: 2/5 M R². Hollow thin sphere: 2/3 M R².
- Solid cylinder about its axis same as disc ½ M R².

Notice: For same M,R, ring has more I than disc because mass farther. So ring harder to rotate.

Force causes translation. What causes rotation? Torque.

Open a door. If you push near hinge, hard to open. Push at handle far from hinge, easy. Also push at 90° to door max effective, push along door zero. So torque depends on distance and angle.

Torque τ = r × F = r F sinθ. r is distance from axis to point where force applied (lever arm), θ angle between r and F. Max when sinθ=1 => θ=90° perpendicular. Unit N m, vector direction by right hand thumb - fingers r to F, thumb is τ direction.

Example: Spanner. Long spanner r large => more torque same force => easier to loosen bolt. That's why we use long handle.

Rotational second law: τ = I α where α = angular acceleration rad/s². Exactly like F=ma. I is rotational mass, α is rotational acceleration.

Angular momentum L is rotational momentum. L = r × p = r × (m v) = I ω for rigid body. Like p=mv, L=I ω. Direction along axis.

Relation: τ = dL/dt rate of change of angular momentum = torque. Analogous to F=dp/dt. If no torque, L constant.

Rotational energy: KE_rot = ½ I ω² analogous to ½ m v². Work done by torque W = τ θ (θ in radians). Power P = τ ω analogous to F v.

See-saw balanced when τ1 = τ2 => r1 W1 = r2 W2. That's why light kid sits far.

For a body to be in complete equilibrium, two conditions: ΣF=0 translational, Στ=0 rotational. Like a book on table at rest both satisfied. If only one zero, it will translate or rotate.

Types: Stable - small disturb returns (ball in bowl, U minimum). Unstable - small disturb falls away (ball on hill top, U maximum). Neutral - stays wherever (ball on flat).

Toppling: When you push a box, it may slide or topple. Toppling occurs when torque due to applied force about edge exceeds torque due to weight. That's why high vehicles topple in sharp turn - COM high, weight torque less.

Now most interesting: Conservation of Angular Momentum. If external torque τ_ext=0, then dL/dt=0 => L = constant. So I1 ω1 = I2 ω2.

Ice skater: Arms stretched I large ω small slow spin. Pulls arms in I decreases (mass closer to axis) so ω increases to keep L same - spins faster. She does work to pull arms in, that work increases rotational KE (KE = L²/2I, so if I decreases KE increases).

Same with diver: Tucks body reduces I spins faster to do somersaults.

Planets: As planet comes closer to sun in elliptical orbit, I about sun decreases? Actually L = m v r sinθ conserved, so speed increases near sun (Kepler second law).

Helicopter: Single rotor would cause body to spin opposite to conserve L, so needs tail rotor to provide counter torque.

Rolling is most common motion: cycle wheel, ball rolling, car tire. It's translation of COM + rotation about COM together.

Pure rolling without slipping condition: v_cm = ω R. Means point of contact instantaneously at rest w.r.t ground. If v_cm > ωR slipping forward like car skidding, if v_cm < ωR spinning like bike wheel stuck in mud.

Why condition? Bottom point velocity = V_cm (forward) + (-ωR) (backward due to rotation) = 0 if V_cm=ωR.

Kinetic energy of rolling: Total KE = Translational ½ M V_cm² + Rotational ½ I_cm ω². Both present. So rolling object has more KE than just sliding same speed because rotation adds.

Using v_cm=ωR, KE = ½ M V² + ½ I V²/R² = ½ M V² (1 + I/MR²) = ½ M V² (1 + K²/R²) where K radius of gyration.

Rolling down incline: Potential loss mgh sin? Actually height h = s sinθ. Energy conservation (if pure rolling static friction does no work - no displacement at contact): m g h = ½ M V² + ½ I ω² = ½ M V² (1+I/MR²). So V = √[2 g h / (1+I/MR²)]. Acceleration a = g sinθ / (1+I/MR²) = g sinθ / (1+K²/R²).

Check: Ring I=MR² => a = g sinθ /2 = 0.5 g sinθ slowest. Disc I=½ MR² => a= (2/3) g sinθ ≈0.66. Solid sphere I=2/5 MR² => a= (5/7) g sinθ ≈0.71 fastest. So solid sphere reaches bottom first, ring last. You can test at home with bangles vs coin vs ball.

Friction in pure rolling is static, does no work (contact point no displacement) but provides torque to rotate. If surface too smooth, slipping happens kinetic friction does work and energy lost as heat.