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 (ω), and angular acceleration (α) about the axis.

Angular position (θ)

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

Angular displacement (Δθ)

Change in angular position: Δθ = θf − θi.

Standard sign convention (planar rotation)

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

Radian

Angle unit defined so that θ = s/r; required for simple relations like s = rθ and W = τΔθ.

Arc-length relation

For a point at radius r, arc length s relates to angle by s = rθ (θ in radians).

Average angular velocity (ωavg)

ωavg = Δθ/Δt.

Instantaneous angular velocity (ω)

ω = dθ/dt.

Average angular acceleration (αavg)

αavg = Δω/Δt.

Instantaneous angular acceleration (α)

α = dω/dt = d²θ/dt².

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: at = rα.

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→θ, v→ω, a→α: ω=ω0+αt; θ=θ0+ω0t+(1/2)αt²; ω²=ω0²+2α(θ−θ0); θ−θ0=(1/2)(ω+ω0)t; θ−θ0=ωt−(1/2)αt².

Torque (τ)

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

Torque magnitude (rFsinφ)

Magnitude of torque: τ = rF sinφ, where φ is the angle between r⃗ (pivot to application point) and F⃗.

Moment arm (lever arm)

Perpendicular distance from the axis to the force’s line of action; torque magnitude can be written τ = F r⊥.

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 (τnet)

Sum of all torques about an axis: τnet = Στi; if τnet = 0 then α = 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 (α=0) while net torque is zero (τnet=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 = Σ mi ri² (ri is distance to axis).

Moment of inertia for continuous objects

For a mass distribution: I = ∫ r² dm.

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 = (1/2)MR².

Solid sphere moment of inertia

About a diameter: I = (2/5)MR².

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 = Icm + Md².

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: τnet = Iα.

Torque from tangential tension on a pulley

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

No-slip constraint (string or rolling contact)

If no slipping occurs at radius R, linear and angular variables relate by v = ωR and a = αR.

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: Krot = (1/2)Iω².

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 = ∫ τ dθ; if τ is constant, W = τΔθ (θ 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: L⃗ = r⃗ × p⃗; magnitude L = r p sinφ.

Angular momentum (rigid body, fixed axis)

For rotation about a fixed axis: L = Iω (about that axis).

Torque–angular momentum relation

Net external torque equals the time rate of change of angular momentum: τ⃗net = dL⃗/dt.

Conservation of angular momentum

If net external torque about a point/axis is zero (τext = 0), angular momentum about that point/axis is constant: Li = Lf.

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 vcm = ωR (and acm = αR).

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.