Rotational Dynamics Formulas to Know for AP Physics C: Mechanics (2025)

1) What You Need to Know

Rotational dynamics is the “torque version” of Newton’s laws: forces cause linear acceleration, and torques cause angular acceleration. On AP Physics C: Mechanics, nearly every rotation problem is some mix of:

• Torque $\tau$ from forces
• Moment of inertia $I$ (how “hard” it is to spin)
• Angular acceleration $\alpha$, angular speed $\omega$
• Angular momentum $\vec L$ and its conservation
• Energy (work/rotational kinetic energy)
• Rolling without slipping constraints

The two core “master equations” you must know:

• Rotational Newton’s 2nd law (fixed axis): $\sum \tau_{\text{ext}} = I\alpha$
• General torque–angular momentum relation: $\sum \vec\tau_{\text{ext}} = \frac{d\vec L}{dt}$

Use rotational dynamics when:

• A rigid body (disk, rod, pulley, wheel) is accelerating angularly.
• Forces act at a distance from an axis (torques matter).
• You need to connect translation + rotation (pulleys, rolling objects).

Big exam idea: Choose an axis strategically. Picking the right torque axis can eliminate unknown forces (e.g., hinge forces) and save you tons of algebra.

2) Step-by-Step Breakdown

A. Standard “Torque + Translation” Recipe (most AP C rotation problems)

1. Choose a coordinate system (sign convention for rotation too: CCW positive is common).
2. Draw a clean FBD for every object that moves (blocks + rotating body).
3. Pick an axis for torques (often the rotation axis). Decide what torques are positive.
4. Write translation equations for each mass: $\sum F = ma$ along the motion direction.
5. Write the rotation equation for the rotating body:
   • Fixed axis: $\sum \tau = I\alpha$
   • Use $\tau = rF\sin\phi$ (or vector $\vec\tau = \vec r \times \vec F$)
6. Add kinematic/constraint relations:
   • No-slip string on pulley: $a = \alpha R$
   • Rolling without slipping: $v_{\text{cm}} = \omega R$ and $a_{\text{cm}} = \alpha R$
7. Solve the system (you usually have as many equations as unknowns).
8. Check limiting cases (e.g., if $I\to 0$ pulley becomes “massless,” does your result match intuition?).

B. Mini Worked Workflow (pulley constraint)

Suppose a mass pulls a string wrapped on a pulley of radius $R$.

• Translational: $mg - T = ma$
• Rotational about pulley axle: $TR = I\alpha$
• No slip: $a = \alpha R$
  Combine: $TR = I\frac{a}{R} \Rightarrow T = \frac{Ia}{R^2}$, then plug into translation.

Decision point: If the axis is fixed and the body is rigid, use $\sum\tau = I\alpha$. If the axis moves or you’re asked about angular momentum conservation, use $\sum\vec\tau_{\text{ext}} = \frac{d\vec L}{dt}$ and/or energy.

3) Key Formulas, Rules & Facts

A. Core Definitions (torque, angular momentum, inertia)

Units & angles:

• $I$ in $\text{kg}\cdot\text{m}^2$, $\tau$ in $\text{N}\cdot\text{m}$, $\omega$ in $\text{rad/s}$, $\alpha$ in $\text{rad/s}^2$.
• Use radians in kinematics/energy. (Degrees break formulas.)

B. Rotational Kinematics (constant $\alpha$)

Linear–angular links (for a point at radius $r$):

• Tangential speed: $v = \omega r$
• Tangential acceleration: $a_t = \alpha r$
• Centripetal acceleration: $a_c = \omega^2 r$

C. Work, Energy, and Power in Rotation

Reminder: Static friction can do zero work in pure rolling (contact point instantaneously at rest), yet it can still provide a torque that changes $\omega$.

D. Rolling Without Slipping (high-yield)

Common moments of inertia ratios $\frac{I_{\text{cm}}}{mR^2}$:

• Solid disk/cylinder: $\tfrac12$
• Hoop/thin ring: $1$
• Solid sphere: $\tfrac25$
• Thin spherical shell: $\tfrac23$

E. Standard Moments of Inertia (know these cold)

Theorems:

• Parallel-axis theorem: $I = I_{\text{cm}} + md^2$ (shift axis by distance $d$)
• Perpendicular-axis theorem (planar lamina): $I_z = I_x + I_y$ (only for flat objects in the $xy$-plane)

F. Angular Impulse & Momentum Conservation

G. Common Torque Setups (fast recognition)

• Force applied tangentially at radius $R$: $\tau = FR$
• Weight on a rod pivoted at one end (COM at $L/2$): $\tau_g = mg\left(\tfrac{L}{2}\right)\sin\theta$ (where $\theta$ is angle between rod and vertical if you define it that way—be consistent)
• Multiple forces: sum torques with sign.

Critical: Torque depends on the perpendicular lever arm $r_\perp$: $\tau = Fr_\perp$.

4) Examples & Applications

Example 1: Block + Massive Pulley (classic AP C)

A block of mass $m$ hangs from a string wrapped around a pulley (radius $R$, inertia $I$). Find acceleration magnitude $a$.

Setup:

• Block: $mg - T = ma$
• Pulley: $TR = I\alpha$
• Constraint: $a = \alpha R$

Key solve:
$TR = I\frac{a}{R} \Rightarrow T = \frac{Ia}{R^2}$
Plug into translation:
$mg - \frac{Ia}{R^2} = ma \Rightarrow a = \frac{mg}{m + \frac{I}{R^2}}$
Insight: Pulley inertia acts like “extra mass” $\frac{I}{R^2}$.

Example 2: Rolling Object Down an Incline

A rigid body (mass $m$, radius $R$, inertia $I_{\text{cm}}$) rolls without slipping down incline angle $\beta$.

Fast result (no need to solve for friction explicitly):
$a = \frac{g\sin\beta}{1 + \frac{I_{\text{cm}}}{mR^2}}$
Ranking speed at bottom (same drop height): smaller $\frac{I}{mR^2}$ wins.

• Solid sphere $\tfrac25$ fastest
• Solid disk $\tfrac12$ next
• Hoop $1$ slowest

Example 3: Door Torque (lever arm + angle trap)

You push on a door at distance $r$ from hinges with force $F$ at angle $\phi$ to the door (in the plane).

Torque magnitude about hinge:
$\tau = rF\sin\phi$
Key insight: Pushing perpendicular to the door $\Rightarrow \sin\phi = 1$ gives max torque.

Common exam twist: same $F$ but different push point: doubling $r$ doubles $\tau$.

Example 4: Angular Momentum Conservation (person on stool)

A person on a frictionless rotating stool pulls arms in, changing inertia from $I_i$ to $I_f$.

If external torque is negligible:
$I_i\omega_i = I_f\omega_f \Rightarrow \omega_f = \frac{I_i}{I_f}\,\omega_i$
Energy is not conserved here (muscles do internal work):
$K_{\text{rot}} = \tfrac12 I\omega^2$ increases when $I$ decreases.

5) Common Mistakes & Traps

1. Mixing up $r$ vs. $r_\perp$ (lever arm).

   • Wrong: using $\tau = rF$ automatically.
   • Right: $\tau = rF\sin\phi = Fr_\perp$. Draw the perpendicular distance to the line of action.

2. Forgetting that torque depends on axis choice.

   • Wrong: computing torque about the wrong point, then wondering why hinge forces appear.
   • Fix: choose an axis that eliminates unknown forces (e.g., about a pivot so pivot forces give zero torque).

3. Using $\sum \tau = I\alpha$ when the axis isn’t fixed / body isn’t a simple rigid rotation.

   • Wrong: applying it blindly in situations with moving axes.
   • Fix: if unsure, fall back to $\sum\vec\tau_{\text{ext}} = \frac{d\vec L}{dt}$ or use energy.

4. Sign errors (CW vs CCW) when summing torques.

   • Wrong: mixing sign conventions between translation and rotation.
   • Fix: declare “CCW positive” (or CW), then stick to it across the problem.

5. Assuming friction always opposes motion (rolling friction confusion).

   • Wrong: claiming static friction must point uphill on an incline.
   • Truth: static friction opposes relative slipping at the contact point. Its direction depends on the tendency to slip.

6. Forgetting the constraint $a = \alpha R$ (strings and rolling).

   • Wrong: solving translation and rotation separately.
   • Fix: write constraints early; they are often the missing equation.

7. Using degrees instead of radians in kinematics/energy.

   • Wrong: plugging $\theta$ in degrees into $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$.
   • Fix: convert to radians or keep everything symbolic.

8. Treating tension as the same on both sides of a massive pulley.

   • Wrong: setting $T_1=T_2$ when pulley has nonzero $I$.
   • Fix: use torque: $(T_1-T_2)R = I\alpha$.

6) Memory Aids & Quick Tricks

7) Quick Review Checklist

• You can write and use $\sum \tau = I\alpha$ (fixed axis) and $\sum\vec\tau = \frac{d\vec L}{dt}$ (general).
• You consistently compute torque using $\tau = rF\sin\phi = Fr_\perp$ with correct signs.
• You know the standard $I$ formulas (disk, hoop, rod, sphere) and can use $I = I_{\text{cm}} + md^2$.
• You remember rotational kinematics (constant $\alpha$): $\omega = \omega_0+\alpha t$, $\omega^2 = \omega_0^2+2\alpha\Delta\theta$.
• You can switch between linear and angular: $v=\omega r$, $a_t=\alpha r$.
• You can do rotational energy: $K_{\text{rot}}=\tfrac12 I\omega^2$ and work by torque $W=\int\tau\,d\theta$, power $P=\tau\omega$.
• For rolling without slipping you immediately write $v_{\text{cm}}=\omega R$ and (if needed) $a_{\text{cm}}=\alpha R$.
• You avoid traps: tension differs across massive pulleys; static friction direction depends on slip tendency.

You’ve got this—if you set up torques cleanly and lock in the constraints, the algebra usually falls into place.