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Rotation

Source lecture(s): SC133 Lec 12

Intuition

Everything you learned about straight-line motion has a spinning twin. Angle replaces position, angular velocity replaces velocity, torque replaces force, moment of inertia replaces mass — and every equation carries over unchanged. Learn the dictionary once and rotation costs you nothing new; the only genuinely fresh idea is that the "mass" of rotation depends on where the mass sits.

The translation dictionary

Linear Rotational Link
position \(x\) angle \(\theta\) (rad) \(s = r\theta\)
velocity \(v\) angular velocity \(\omega = d\theta/dt\) \(v = r\omega\)
acceleration \(a\) angular acceleration \(\alpha = d\omega/dt\) \(a_t = r\alpha\)
mass \(m\) moment of inertia \(I\) see below
force \(F\) torque \(\tau\) \(\tau = rF\sin\phi\)
\(\sum F = ma\) \(\sum\tau = I\alpha\) Newton II, rotated
\(K = \tfrac12 mv^2\) \(K = \tfrac12 I\omega^2\)
\(p = mv\) \(L = I\omega\) angular momentum

The constant-\(\alpha\) kinematics come along for free: \(\omega = \omega_0 + \alpha t\), \(\theta = \omega_0 t + \tfrac12\alpha t^2\), \(\omega^2 = \omega_0^2 + 2\alpha\Delta\theta\) — the kinematic equations with letters swapped.

Radians, or the dictionary breaks

The linking relations \(s = r\theta\), \(v = r\omega\) hold only in radians (the definition of angle as arc/radius). One revolution \(= 2\pi\) rad; \(1\,\text{rpm} = 2\pi/60\,\text{rad/s}\).

Every point of a rigid body

All points share the same \(\omega\) and \(\alpha\); each point's linear speed grows with radius, \(v = r\omega\). A point at radius \(r\) also has centripetal acceleration \(a_c = \omega^2 r\) toward the axis — circular motion applied point by point.

Worked example: spinning up a flywheel

A flywheel with \(I = 2\,\text{kg·m}^2\) starts at rest under constant torque \(\tau = 10\,\text{N·m}\) for 6 s. Find \(\omega\), the angle turned, and the kinetic energy.

\(\alpha = \tau/I = 5\,\text{rad/s}^2\); \(\omega = \alpha t = 30\,\text{rad/s}\); \(\theta = \tfrac12\alpha t^2 = 90\,\text{rad}\) (≈14.3 rev); \(K = \tfrac12 I\omega^2 = 900\,\text{J}\) — which equals the work \(\tau\theta = 10\times90\) ✓.

Common mistakes

  • Degrees in the linking formulas. \(v = r\omega\) with \(\omega\) in deg/s is meaningless; convert first.
  • Confusing \(a_t\) and \(a_c\). Tangential acceleration (\(r\alpha\)) changes speed; centripetal (\(\omega^2 r\)) changes direction; both may coexist.
  • Treating \(\omega\) as the same for different bodies connected by belts: belts share linear speed at the rim (\(v = r\omega\)), so different radii spin at different \(\omega\).

Knowledge graph position

Prerequisites: Kinematics, Newton's laws, Circular motion. Leads to: Moment of inertia, Torque, Rolling & angular momentum.

Quiz

Q1 (computational). A hard disk spins at 7200 rpm. Angular velocity in rad/s, and linear speed at radius 4 cm?

Answer

\(\omega = 7200\times2\pi/60 = 754\,\text{rad/s}\); \(v = r\omega = 0.04\times754 \approx 30\,\text{m/s}\) — over 100 km/h at the rim.

Q2 (conceptual). Two children sit on a merry-go-round, one near the center, one at the edge. Compare their angular and linear velocities.

Answer

Same \(\omega\) (rigid body), but the edge child moves faster linearly (\(v = r\omega\)) — and needs more centripetal force to hold on.

Q3 (multiple choice). Doubling a flywheel's angular speed multiplies its kinetic energy by: (a) 2 (b) 4 (c) \(\sqrt2\)

Answer

(b). \(K = \tfrac12 I\omega^2 \propto \omega^2\) — exactly like \(v^2\) for linear motion; the dictionary never lies.