Unit 5: Rotation

Rotational motion

Motion of an object turning about an axis; rotational analogs exist for position (angle), velocity (angular speed), and acceleration (angular acceleration).

Rigid-body rotation

Rotation where the object does not deform; all points on the body share the same angular displacement, angular velocity ($\omega$), and angular acceleration ($\alpha$) about the axis.

Angular position (θ)

The rotational “position” variable describing orientation, measured in radians.

Angular displacement (Δθ)

Change in angular position: $\triangle \theta = \theta_f - \theta_i.$.

Standard sign convention (planar rotation)

Counterclockwise is positive (when looking along the positive axis direction); clockwise is negative.

Radian

Angle unit defined so that $\theta = \frac{s}{r}$; required for simple relations like $s = r\theta$ and $W = \tau \Delta \theta$.

Arc-length relation

For a point at radius $r$, arc length $s$ relates to angle by $s = r\theta$ ( heta in radians).

Average angular velocity (ωavg)

$\bar{\boldsymbol{\theta}} = \frac{\triangle \theta}{\triangle t}$.

Instantaneous angular velocity (ω)

$\omega = \frac{d\theta}{dt}$.

Average angular acceleration ($\alpha_{avg}$)

$\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$.

Instantaneous angular acceleration ($\alpha$)

$$\boldsymbol{\beta} = \frac{d\boldsymbol{\theta}}{dt} = \frac{d^2\boldsymbol{\theta}}{dt^2}.$$

Tangential (linear) speed in rotation

Linear speed of a point at radius r: v = rω.

Tangential acceleration (at)

Acceleration that changes speed along the circular path: $a_t = r\alpha$.

Centripetal (radial) acceleration (ac)

Acceleration toward the axis that changes direction of velocity: ac = rω² (not rα²).

Big Five rotational kinematics equations (constant α)

Analogous to linear kinematics with $x \rightarrow \theta, v \rightarrow \boldsymbol{\beta}, a \rightarrow \boldsymbol{\beta}$: $\boldsymbol{\beta} = \boldsymbol{\beta_0} + \boldsymbol{\beta}t; \theta = \theta_0 + \boldsymbol{\beta_0} t + \frac{1}{2} \boldsymbol{\beta} t^2; \boldsymbol{\beta}^2 = \boldsymbol{\beta_0}^2 + 2\boldsymbol{\beta}(\theta - \theta_0); \theta - \theta_0 = \frac{1}{2}(\boldsymbol{\beta} + \boldsymbol{\beta_0})t; \theta - \theta_0 = \boldsymbol{\beta}t - \frac{1}{2} \boldsymbol{\beta} t^2.$

Torque (τ)

The rotational effectiveness of a force about a chosen axis; vector definition τ⃗ = r⃗ × F⃗.

Torque magnitude (rFsinφ)

Magnitude of torque: $\tau = rF \sin \phi$, where $\phi$ is the angle between $\vec{r}$ (pivot to application point) and $\vec{F}$.

Moment arm (lever arm)

Torque magnitude can be written $\tau = F \thinspace r_{\bot}$.

Right-hand rule (torque direction)

Torque direction is along the rotation axis, perpendicular to the plane of rotation, determined by r⃗ × F⃗ (curl fingers from r⃗ to F⃗).

Net torque ($\tau_{net}$)

Sum of all torques about an axis: $\tau_{net} = \sum \tau_i$; if $\tau_{net} = 0$ then $\alpha = 0$.

Static equilibrium (rotation + translation)

Object at rest with no linear or angular acceleration: ΣFx=0, ΣFy=0, and Στ=0 about any chosen point.

Dynamic rotational equilibrium

Object rotates with constant angular velocity ($\alpha=0$) while net torque is zero ($\tau_{net}=0$).

Couple

A pair of equal and opposite forces separated by a distance that produces rotation (nonzero net torque) even if net force is zero.

Moment of inertia (I)

Rotational analog of mass; measures resistance to angular acceleration and depends on mass distribution relative to the axis (units: kg·m²).

Moment of inertia for point masses

For discrete masses: $I = \sum m_i r_i^2$ (where $r_i$ is distance to axis).

Moment of inertia for continuous objects

For a mass distribution: $I = \frac{\text{d}m}{r^2}.$.

Thin hoop (ring) moment of inertia

About central axis perpendicular to the plane: I = MR².

Solid disk (or solid cylinder) moment of inertia

About central axis: $I = \frac{1}{2}MR^2$.

Solid sphere moment of inertia

About a diameter: $I = \frac{2}{5}MR^2$.

Thin rod moment of inertia (about center)

Axis through center, perpendicular to rod: I = (1/12)ML².

Thin rod moment of inertia (about one end)

Axis through one end, perpendicular to rod: I = (1/3)ML².

Parallel-axis theorem

Shifts MOI from center-of-mass axis to a parallel axis a distance $d$ away: $I = I_{cm} + Md^2$.

Perpendicular-axis theorem

For a flat lamina in the xy-plane about perpendicular axes through the same point: Iz = Ix + Iy.

Rotational Newton’s second law

For fixed-axis rigid-body rotation: $\tau_{net} = I\alpha$.

Torque from tangential tension on a pulley

If a string pulls tangentially at radius $R$ with tension $T$, the torque magnitude is $\tau = TR$.

No-slip constraint (string or rolling contact)

If no slipping occurs at radius $R$, linear and angular variables relate by $v = \omega R$ and $a = \alpha R$.

Hanging mass + solid-disk pulley acceleration

For mass m on a string around a solid disk pulley (mass M), no slip: a = mg/(m + (1/2)M).

Rotational kinetic energy

Energy of rotation about an axis: $K_{rot} = \frac{1}{2}I\omega^2$.

Total kinetic energy (translation + rotation)

For a rigid body: K = (1/2)Mvcm² + (1/2)Icm ω² (important for rolling).

Work done by a torque

Rotational work: $W = \int \tau \, d\theta$; if $\tau$ is constant, $W = \tau \Delta \theta$ ( heta in radians).

Rotational power

Rate of doing rotational work: P = τω.

Mechanical energy conservation with rotation

When only conservative forces do work: Ki + Ui = Kf + Uf, where K may include both translational and rotational kinetic energy.

Angular momentum (particle)

About an origin: $\boldsymbol{L} = \boldsymbol{r} \times \boldsymbol{p}$; magnitude $L = rp \thinspace \sin \thinspace \theta$.

Angular momentum (rigid body, fixed axis)

For rotation about a fixed axis: $L = I\omega$ (about that axis).

Torque–angular momentum relation

Net external torque equals the time rate of change of angular momentum: $\boldsymbol{\tau}_{\text{net}} = \frac{d\boldsymbol{L}}{dt}.$.

Conservation of angular momentum

If net external torque about a point/axis is zero ($\tau_{ext} = 0$), angular momentum about that point/axis is constant: $L_i = L_f$.

Rotational kinetic energy with fixed angular momentum

If L is constant, K = L²/(2I); decreasing I increases K (internal work can change energy even when L is conserved).

Rolling without slipping

Pure rolling where the contact point is instantaneously at rest relative to the ground; the center-of-mass speed satisfies $v_{cm} = \omega R$ (and $a_{cm} = \alpha R$).

Tipping condition (stability)

An object is stable as long as the line of action of its weight falls within the base of support; at the tipping point, the normal force effectively acts at the edge and torques about that edge balance.

Inelastic rotational collision (sticking to a rotating object)

During a brief sticking collision, angular momentum about the chosen axis may be conserved (if external torque is negligible) but mechanical energy is not; e.g., mvR = (Idisk + mR²)ω.