Newton's Second Law in rotational form states that the angular acceleration of a rigid system equals the net torque divided by the system's rotational inertia:
tau_net = I * alpha
Equivalently: alpha = tau_net / I. This is the rotational analog of F_net = ma in every meaningful way. Net torque is the rotational cause. Angular acceleration is the rotational effect. Rotational inertia is the resistance between them.
Linear: F_net = m * a → a = F_net / m
Rotational: tau_net = I * alpha → alpha = tau_net / I
🔑Angular velocity changes only when net torque is nonzero. If Στ = 0 → alpha = 0 → omega is constant (5.5 rotational equilibrium). If Στ ≠ 0 → alpha ≠ 0 → omega is changing (this lesson). These are the two sides of the same coin.
5.6.A.2Concept
Proportional Relationships
From alpha = tau_net / I, two functional relationships follow directly. The AP exam tests these as reasoning questions — predicting how alpha changes when tau or I changes by a given factor.
Direct proportionality
alpha ∝ tau_net (fixed I)
Double the net torque → double the angular acceleration. Apply twice the torque to the same disk and it spins up twice as fast.
Inverse proportionality
alpha ∝ 1/I (fixed tau_net)
Double the rotational inertia → half the angular acceleration. Switch to a disk twice as heavy and it responds half as vigorously to the same torque.
💡AP exam phrasing to watch for: "If the net torque is tripled while the rotational inertia is halved, what happens to the angular acceleration?" Answer: alpha changes by a factor of 3 × 2 = 6. Multiply the torque factor by the inverse of the inertia factor.
5.6.A.3ConceptMath
Hoop vs. Disk — Same Torque, Different Response
A hoop and a solid disk of equal mass M and radius R have different rotational inertias: I_hoop = MR² and I_disk = ½MR². Apply the same net torque to each and the disk accelerates twice as fast.
Apply a torque to a disk or hoop of the same mass and radius. Watch how shape (mass distribution) changes I — and therefore alpha. Same torque, same mass, same size: different response.
Net torque tau12.0 N·m
Mass M2.0 kg
Radius R0.4 m
I (disk)
0.160 kg·m²
alpha = tau/I
75.00 rad/s²
a_rim = R*alpha
30.00 m/s²
Angular acceleration (max scale = 60 rad/s²)
75.0 rad/s²
Switch between disk and hoop — same tau, M, and R but the hoop has twice the I of the disk, so it gets exactly half the angular acceleration. This is the r² effect from 5.4 expressed as a real dynamic consequence.
ExampleWorked Example — Torque on a Disk vs. Hoop
A 3 kg, 0.5 m radius disk and a 3 kg, 0.5 m radius hoop each have a 6 N·m net torque applied. Find alpha for each and the rim's linear acceleration.
5.6.A.4Math
Combined Linear and Rotational Systems
The most tested 5.6 problem type: a system where linear and rotational motion are coupled. A mass falls and unwinds a string from a pulley. A block on a ramp drives a disk. These problems require two separate Newton's Second Law equations — one linear, one rotational — connected by the kinematic constraint a = r*alpha.
🔑The three-step framework:(1) Write ΣF = ma for each linearly accelerating object separately. (2) Write Στ = I*alpha for each rotating object separately. (3) Use a = r*alpha to connect the two analyses and solve the system.
ExampleGuided Example — Mass on a Pulley
A 2 kg mass hangs from a string wrapped around a solid disk pulley (mass 4 kg, radius 0.3 m). The system starts from rest. Find the linear acceleration of the mass and the angular acceleration of the pulley. (g = 10 m/s²)
Step 1 — Identify the two systems
System 1: the hanging mass (linear motion, downward). System 2: the disk pulley (rotational motion about its axle).
5.6.A.5Concept
AP Problem Strategy — Full Checklist
Every combined linear+rotational FRQ on the AP exam follows the same structure. Use this checklist as your setup routine before writing any equations.
1Draw a diagram. Label the axis of rotation, all forces, and their points of application.
2Identify each object's motion type: linear (ΣF = ma) or rotational (Στ = I*alpha) or both.
3Write a separate Newton's Second Law equation for each object. Do not mix them.
4Write the kinematic constraint: a = r*alpha. State clearly which r you are using.
5Substitute the constraint into your equations to eliminate alpha (or a). Solve for the remaining unknown.
6Check units, check sign consistency, check that your answer is physically reasonable.
⚠️The most common error on combined problems: using the same equation for both objects. The hanging mass needs ΣF = ma (linear). The pulley needs Στ = I*alpha (rotational). These are different equations for different physics — they must be written separately before being connected.
Unit 5 complete. Progress Check 5 covers all six lessons. Combined linear+rotational FRQs require two separate equation setups, explicit use of a = r*alpha as the connecting constraint, and a clearly labeled diagram.