Lesson 11: Rotational Mechanics

What are the rotational variables?

θ = angular position (rad), ω = angular velocity (rad/s), α = angular acceleration (rad/s²). Positive direction = counterclockwise.

What is average angular velocity?

ω = Δθ/Δt — Change in angular position over time. Units: rad/s

What is average angular acceleration?

α = Δω/Δt — Change in angular velocity over time. Units: rad/s²

What are the kinematic equations for constant angular acceleration?

ω = ω₀ + αt | θ = θ₀ + ω₀t + ½αt² | ω² = ω₀² + 2αΔθ — Direct analogs of linear kinematics

What is the relationship between arc length and angle?

s = rθ — s = arc length, r = radius, θ must be in radians

What is the relationship between tangential velocity and angular velocity?

v_t = rω — Tangential speed equals radius times angular velocity

What is the relationship between tangential acceleration and angular acceleration?

a_t = rα — Tangential acceleration equals radius times angular acceleration

What is centripetal acceleration in terms of angular velocity?

a_c = rω² = v²/r — Always directed toward the center of rotation

What is the total acceleration of a point on a rotating object?

a_total = √(a_t² + a_c²) — Vector sum of tangential and centripetal components

What is moment of inertia?

I = Σmᵢrᵢ² — Sum of each mass element times its distance from the rotation axis squared. Units: kg·m²

What is rotational kinetic energy?

KE_rot = ½Iω² — Rotational analog of ½mv²; I plays the role of mass

What is the moment of inertia of a solid disk or cylinder about its central axis?

I = ½MR² — M = total mass, R = radius

What is the moment of inertia of a solid sphere about its central axis?

I = ⅖MR²

What is the moment of inertia of a hollow sphere about its central axis?

I = ⅔MR²

What is the moment of inertia of a thin rod about its center?

I = (1/12)ML² — L = length of rod

What is the moment of inertia of a thin rod about its end?

I = (1/3)ML²

What is the moment of inertia of a hoop or thin ring about its central axis?

I = MR²

What is the Parallel Axis Theorem?

I = I_cm + Md² — Allows calculation of I about any axis parallel to the CM axis; d = distance between axes

What is torque?

τ = rF sinθ = r⊥F — The rotational effect of a force; θ = angle between r and F. Units: N·m

What is the lever arm?

r⊥ = r sinθ — The perpendicular distance from the rotation axis to the line of action of the force

What is Newton's Second Law for rotation?

Στ = Iα — Net torque equals moment of inertia times angular acceleration; rotational analog of F = ma

What is the work done by a torque?

W = τΔθ — Work equals torque times angular displacement (θ in radians)

What is power in rotational motion?

P = τω — Rotational analog of P = Fv

What is the work-energy theorem for rotation?

W_net = ΔKE_rot = ½Iω_f² - ½Iω_i²

What is rolling motion without slipping?

The contact point has zero velocity. Translational and rotational motion are linked: v_cm = Rω and a_cm = Rα

What is the total kinetic energy of a rolling object?

KE_total = ½mv_cm² + ½Iω² — Sum of translational KE of the CM and rotational KE about the CM

What is angular momentum of a particle?

L = r × p = mvr sinθ — Cross product of position and linear momentum. Units: kg·m²/s

What is angular momentum of a rigid body?

L = Iω — Rotational analog of p = mv

What is Newton's Second Law in terms of angular momentum?

Στ = dL/dt — Net torque equals the rate of change of angular momentum

What is conservation of angular momentum?

L_i = L_f when Στ_ext = 0 — Total angular momentum is conserved when net external torque is zero

Give an example of conservation of angular momentum.

A spinning figure skater pulls their arms in (↓r → ↓I) causing ω to increase to keep L = Iω constant

What is precession of a gyroscope?

When a spinning gyroscope is acted on by a gravitational torque, its spin axis rotates (precesses) around the vertical axis instead of falling

What is the precession rate of a gyroscope?

Ω = τ/L = Mgr/Iω — Precession rate; faster spin = slower precession