Unit 5 — Torque and Rotational Dynamics

Rotational Kinematics

Rotation is described with angular quantities, measured in radians. One full revolution is 2π rad = 360°.

For constant angular acceleration, the equations mirror the linear ones:

ω  = ω₀ + α·t
Δθ = ω₀·t + ½·α·t²
ω² = ω₀² + 2·α·Δθ

• Choose a positive rotation direction (usually counterclockwise) and use signs, just as with linear motion.
• If ω and α have the same sign, the object spins faster; opposite signs, it slows down.
• Period and frequency: T = 2π/ω and f = 1/T. To convert: 1 rpm = 2π/60 rad/s.
• Graphs work as before: the slope of θ–t is ω, the slope of ω–t is α, and the area under ω–t is Δθ.

Connecting Linear and Rotational Motion

Every point on a rigid rotating object has the same ω and α, but points farther from the axis move faster along a longer circle.

s   = r·θ            (arc length)
v_t = r·ω            (tangential speed)
a_t = r·α            (tangential acceleration)
a_c = ω²·r = v²/r    (centripetal acceleration, toward the axis)

• The tangential quantities (vt, at) point along the circle; the centripetal acceleration points toward the axis. A point on a spinning object can have both.
• These relations require angles in radians.
• Rolling without slipping: the center moves at v = ωr, and the distance rolled equals the arc length turned, d = rθ.

Torque

Torque measures how effectively a force causes or changes rotation about a chosen axis (pivot).

τ = r · F · sin θ  =  r⊥ · F        units: N·m

• r is the distance from the axis to where the force is applied; θ is the angle between the vector r (from the axis to the point of application) and the force.
• Only the force component perpendicular to r produces torque. The perpendicular distance from the axis to the force's line of action is the lever arm, r⊥.
• Torque is greatest when the force is perpendicular to r (θ = 90°). A force along the line through the axis (θ = 0° or 180°) gives zero torque.
• Sign convention: counterclockwise torque is positive, clockwise is negative. The net torque is the signed sum Στ.
• Torque depends on the axis chosen. Weight acts at the object's center of mass.

Rotational Inertia

Rotational inertia (moment of inertia), I, is an object's resistance to changes in its rotation. It plays the role of mass in rotational motion.

I = Σ m·r²          units: kg·m²

• Each piece of mass contributes m·r², where r is its distance from the axis of rotation.
• I depends on the total mass, how the mass is distributed, and the choice of axis. The same object has different I about different axes.
• Mass farther from the axis contributes far more (because of r²). Moving mass outward increases I.
• A single point mass on a light rod or string: I = mr².

Rotational Equilibrium and Newton's First Law in Rotational Form

The rotational form of Newton's first law: an object keeps rotating at constant angular velocity (or stays non-rotating) unless a net external torque acts on it.

Rotational equilibrium:   Στ = 0     (α = 0)
Static equilibrium:       ΣF = 0  and  Στ = 0

• In static equilibrium the object is not translating and not rotating, so both the net force and the net torque are zero.
• You may choose any point as the pivot when writing Στ = 0. Pick the point where an unknown force acts, since that force then has zero torque and drops out.
• A uniform object's weight acts at its center of mass, at its geometric center.
• Balanced torques: τcw = τccw, which is the principle of a seesaw or balance.

Steps: draw an FBD, choose a pivot, write each torque with its sign, set Στ = 0 and ΣF = 0, then solve.

Newton's Second Law in Rotational Form

A net torque on an object produces an angular acceleration that is proportional to the torque and inversely proportional to the rotational inertia.

α = Στ / I        or        Στ = I·α

• The same torque gives a smaller α to an object with a larger I.
• Use the same axis for both τ and I, and use signs for direction (α has the sign of the net torque).
• For problems that combine rotation and translation (e.g. a hanging mass on a pulley), write ΣF = ma for the mass, Στ = Iα for the pulley, and link them with a = αr.