The Wheel of Motion

NODE 01

Why Rotation Exists

VS

• Translational vs Rotational motion.
• Particle vs Rigid-body motion.
• Fixed axis, internal/external axes.

NODE 02

Angular Position

• Reference direction, initial/final pos.
• Clockwise/Anticlockwise convention.
• π rad = 180°, 1 rev = 2π rad.

NODE 03

Angular Displacement

• Symbol θ. Rotation through an angle.
• Positive and negative displacement.
• Relation to arc length: s = rθ.

NODE 04

Angular Velocity

• Symbol ω. Rate of change of θ (dθ/dt).
• Radians per second.
• Uniform: ω = 2π/T = 2πf.

NODE 05

Angular Acceleration

• Symbol α. Rate of change of ω (dω/dt).
• Uniform angular acceleration/deceleration.
• Rotational counterpart to a = dv/dt.

NODE 06

Linear-Rot Dictionary

Pos (x)↔Ang Pos (θ)

Vel (v)↔Ang Vel (ω)

Mass (m)↔Mom Iner (I)

• Rotation is a new language for motion.
• Force ↔ Torque, Momentum ↔ Ang Mom.

NODE 07

Arc Length

• s = rθ (when θ is in radians).
• Connects linear distance to angular motion.
• Crucial for wheels, tracks, gears.

NODE 08

Tangential Velocity

• v = rω.
• Same ω for all points on rigid body.
• Larger radius → larger tangential speed.

NODE 09

Tangential Accel

• at = rα.
• Direction tangent to path.
• Changes the speed of rotation.

NODE 10

Centripetal Accel

• ac = v²/r = ω²r.
• Direction: toward the centre.
• Changes the direction of velocity.

NODE 11

Centripetal Force

• Fc = mv²/r = mω²r.
• Not a new mysterious force.
• It is the NET inward force required.

NODE 12

Period & Frequency

• Period (T): Time for one revolution.
• Frequency (f): Revs per second (1/T).
• ω = 2πf.

NODE 13

Uniform Circular

• Constant speed/angular velocity.
• Changing velocity direction.
• Satellite orbits, ferris wheels.

NODE 14

Non-Uniform Circ

• Both speed and direction change.
• Has Radial (ar) + Tangential (at) accel.
• Total a = √(ar² + at²).

NODE 15

Torque

• Force producing rotation.
• τ = rF sinθ.
• Force changes translation, Torque changes rotation.

NODE 16

Lever / Moment Arm

• Perpendicular distance from axis to force line.
• τ = F × perpendicular distance.
• Door handles, wrenches, seesaws.

NODE 17

Couples

• Two equal, opposite, parallel forces.
• Zero resultant force, non-zero turning effect.
• Steering wheels, turning knobs.

NODE 18

Rotational Equilib

• Translational eq: ΣF = 0.
• Rotational eq: Στ = 0.
• Complete static eq requires both.

NODE 19

Center of Mass

• Average position weighted by mass.
• Symmetry in uniform bodies.
• x_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂).

NODE 20

Stability

↻

• Stable, Unstable, Neutral.
• Depends on base of support & COM height.
• Line of action of weight determines toppling.

NODE 21

Moment of Inertia

• Rotational counterpart to mass.
• Measures resistance to angular acceleration.
• I = Σmr². Mass further from axis contributes more.

NODE 22

Standard MOI

• Point mass, Thin rod, Ring, Disc.
• Solid/Hollow cylinders and spheres.
• MOI depends heavily on the chosen axis.

NODE 23

Parallel Axis Thm

• I = I_cm + Md².
• Relates MOI through COM to a parallel axis at dist d.

NODE 24

Perpendicular Axis

• Iz = Ix + Iy.
• Applies only to suitable planar bodies (lamina).

NODE 25

Rotational Dynamics

τ=Iα

• Rotational counterpart to Newton's 2nd law.
• Net torque = Moment of Inertia × Ang Accel.
• A major destination concept.

NODE 26

Rotational Work

• Torque acting through angular displacement.
• W = τθ (constant torque).
• W = ∫τ dθ (general form).

NODE 27

Rotational Power

• P = τω.
• Extremely useful for motors, engines, turbines.

NODE 28

Rotational KE

K = ½Iω2

• Rotating rigid body: K_rot = ½Iω².
• Compare to K_trans = ½mv².

NODE 29

Rolling Motion

• Translation + Rotation.
• Pure rolling: v = rω (no slipping).
• Total KE = ½Mv² + ½Iω².

NODE 30

Angular Momentum

L = Iω

• Rotational equivalent of linear momentum.
• Rigid body: L = Iω. Particle: L = r × p.

NODE 31

Cons of Ang Mom

• If external torque is negligible, L = constant.
• I₁ω₁ = I₂ω₂.
• Figure skaters, collapsing stars.

NODE 32

Torque-L Connection

τ=dL/dt

• The bridge: Torque is the rate of change of L.
• Mirrors Force = dp/dt.

NODE 33

Rotational Collision

• Angular impulse.
• Conservation of angular momentum.
• Coupled rotating bodies.

NODE 34

Real Machines

• Gears, wheels, flywheels, turbines, motors.
• Rotational equations describe actual civilization.

NODE 35

Gear Systems

• Gear radius, angular speed, torque transfer.
• Direction reversal, mechanical advantage.

NODE 36

Pulleys & Rotation

• Rope tension, rotational inertia, acceleration.
• Integrates Realms 03, 04, and 05.

NODE 37

Experimental Wheel

• Determine ω, centripetal force, torque.
• Find COM, moment of inertia, rolling obj.

NODE 38

Misconceptions

X

• "Constant ang speed = zero accel" → False.
• "Torque is just force" → False.
• "Centripetal force points outward" → False.

NODE 39

Solving Engine

• Identify Axis → Draw Forces → Lever Arms.
• Calc Torques → Apply Στ = Iα or ΣF = ma.

NODE 40

Grand Synthesis

• Translation has F, m, p, and KE_trans.
• Rotation has τ, I, L, and KE_rot.
• Perfect mathematical symmetry.