AP Physics C: Mechanics Unit 5 Cheatsheet: Torque and Rotational Dynamics

Angular displacement, velocity, acceleration:

• Δθ = θ − θ₀ (radians)
• ω = dθ/dt
• α = dω/dt

Constant angular acceleration (parallel to linear 1D):

• ω = ω₀ + αt
• θ = θ₀ + ω₀t + ½αt²
• ω² = ω₀² + 2α(θ − θ₀)

Sign convention: One rotational direction (clockwise or counterclockwise) is positive; the other is negative.

Key insight: All points on a rigid system share the same ω and α about a given axis.

Definition: Resistance to changes in rotation; depends on mass and its distance from the axis.

Point mass: I = mr²

System of masses: I_tot = Σm_ir_i²

Continuous solid: I = ∫r² dm

Key principle: Mass farther from axis → larger I. A hoop has more I than a solid disk of equal mass and radius.

Parallel axis theorem: I′ = I_cm + Md², where d is the distance from the center of mass to the new axis.

Minimum I: Occurs when the axis passes through the center of mass.

Equation: α_sys = Στ / I_sys = τ_net / I_sys

Meaning:

• Angular acceleration is directly proportional to net torque.
• Angular acceleration is inversely proportional to rotational inertia.
• Direction of α is the same as direction of τ_net.

When net torque ≠ 0: Angular velocity changes.

Solving dynamics problems:

• Identify all torques about the chosen axis.
• Calculate I about that axis.
• Use α = τ_net / I to find angular acceleration.
• May need separate linear analysis for the center of mass.