Physics C: Mechanics · 2024-2027 College Board CED · 18 min read · Updated 2026-05-11
Rotation and Angular Momentum — AP Physics C: Mechanics
AP Physics C: Mechanics · 2024-2027 College Board CED · 18 min read
1. Rotational Kinematics with Calculus★★☆☆☆⏱ 4 min
All rotational quantities are directly analogous to 1D linear kinematics. Angular position $\theta(t)$ is measured in radians from a fixed reference axis, with the following relationships:
For a point at perpendicular distance $r$ from the rotation axis, tangential linear quantities are related to angular quantities by $s = r\theta$, $v_t = r\omega$, $a_t = r\alpha$. Centripetal acceleration pointing toward the axis is $a_c = r\omega^2 = \frac{v_t^2}{r}$. For variable acceleration, integrate to find unknown quantities:
Exam tip: Examiners often give non-constant angular acceleration functions, so never default to constant-acceleration kinematic equations unless explicitly told acceleration is fixed.
2. Moment of Inertia★★★☆☆⏱ 4 min
For discrete point masses, moment of inertia is calculated as $I = \sum m_i r_i^2$, where $r_i$ is the perpendicular distance of mass $m_i$ from the rotation axis. For continuous rigid bodies, the sum becomes an integral:
I = \int r^2 dm
To set up the integral, express the infinitesimal mass $dm$ using density: $dm = \lambda dx$ (1D objects), $dm = \sigma dA$ (2D objects), or $dm = \rho dV$ (3D objects). The parallel axis theorem simplifies calculations for axes offset from the center of mass:
I = I_{cm} + Md^2
3. Torque and Rotational Dynamics★★★☆☆⏱ 3 min
The magnitude of torque from a force applied at distance $r$ from the axis is:
\tau = rF\sin\phi
where $\phi$ is the angle between $\vec{r}$ (from axis to force application point) and $\vec{F}$.
Newton's second law for rotation is directly analogous to $F_{net}=ma$ for linear motion:
\tau_{net} = I\alpha
This relationship only holds for rigid bodies with constant moment of inertia rotating around a fixed axis.
4. Angular Momentum and Conservation★★★☆☆⏱ 3 min
For a point mass, angular momentum is $\vec{L} = \vec{r} \times \vec{p}$, with magnitude $L = rmv\sin\phi$. For a rigid body rotating around a fixed axis, this simplifies to:
L = I\omega
The general form of Newton's second law for rotation is $\tau_{net} = \frac{dL}{dt}$. If net external torque is zero, total angular momentum is conserved:
L_{initial} = L_{final}
Conservation holds even when moment of inertia changes, as angular velocity adjusts to keep total $L$ constant.
5. Rolling Motion★★★★☆⏱ 4 min
Rolling motion is a combination of translational motion of the center of mass (CM) and rotational motion around the CM. For the most common tested case, rolling without slipping, the following key relations hold:
v_{cm} = R\omega \quad \quad a_{cm} = R\alpha
where $R$ is the radius of the rolling object. Total kinetic energy for a rolling object is the sum of translational KE of the CM and rotational KE around the CM: