AP Physics 1: Rotational Dynamics

Rotational dynamics is where AP Physics 1 stops feeling like “motion in a straight line” and becomes truly three-dimensional. Instead of forces causing acceleration, you analyze torques causing angular acceleration. Instead of mass resisting changes in velocity, moment of inertia resists changes in rotational speed. The payoff is big: with a small set of rotational analogs to linear motion, you can explain everything from why a door opens easiest at the handle to why a figure skater spins faster when pulling in their arms.

This article lays out the core ideas you need for torque, rotational kinematics, moment of inertia, angular momentum, and rotational energy, including the parallel axis theorem and conservation of angular momentum.

The rotational analogs that organize the whole unit

AP Physics 1 leans heavily on parallels between linear and rotational motion. You will see the same relationships appear with different symbols:

• Position $x$ becomes angle $\theta$ (radians)
• Velocity $v$ becomes angular velocity $\omega$
• Acceleration $a$ becomes angular acceleration $\alpha$
• Force $F$ becomes torque $\tau$
• Mass $m$ becomes moment of inertia $I$

When the rotation is about a fixed axis, the dynamics mirror Newton’s second law:

$${\tau }_{\text{net}}=I\alpha$$

This is not a separate law from $F_{\text{net}}=ma$. It is the same idea applied to rotational motion: net “twisting effect” produces angular acceleration, and the resistance to that acceleration is $I$.

Torque: what actually makes things start rotating

Torque is the rotational effect of a force applied at some distance from an axis. In magnitude form,

$$\tau =rF\sin \varphi$$

where $r$ is the distance from the axis to the point of application, and $\varphi$ is the angle between the position vector and the force. The $\sin \varphi$ term matters because only the component of the force perpendicular to the radius produces rotation.

Practical meaning and common AP traps

• Longer lever arm, bigger torque: Pushing on the doorknob produces more torque than pushing near the hinges because $r$ is larger.
• Direction matters: Pushing straight toward the hinge line (parallel to $r$) can produce almost no rotation because $\sin \varphi \approx 0$.
• Sign conventions: AP problems often treat counterclockwise torque as positive and clockwise as negative. Consistency is more important than the choice.

Static equilibrium and the bridge to dynamics

Even in rotational dynamics, you often begin with equilibrium reasoning. For an object not accelerating,

• Translational equilibrium: $\sum F=0$
• Rotational equilibrium: $\sum \tau =0$

This becomes especially important in systems like beams, ladders, and seesaws, where you may need to locate forces before analyzing motion.

Rotational kinematics: the “constant acceleration” equations return

When angular acceleration is constant, the familiar kinematics equations reappear:

$$\omega ={\omega }_0+\alpha t$$ $$\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$$ $${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$$

Angles in physics are measured in radians, which is critical because radians make relationships between linear and angular quantities simple.

Connecting linear and angular motion

For a point a distance $r$ from the axis:

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

These links show up constantly in rolling motion problems and in any scenario where something rotates while points on it also move through space.

Moment of inertia: rotational “mass” depends on shape

Moment of inertia $I$ tells you how hard it is to change an object’s rotational motion. The key idea is distribution: mass farther from the axis contributes much more.

For point masses, the definition is:

$$I=\sum m_i{r}_i^2$$

That $r^2$ dependence is the reason a hollow cylinder can have a larger moment of inertia than a solid cylinder of the same mass and radius.

The parallel axis theorem (a must-know tool)

Many AP problems involve rotation about an axis that is not through the center of mass. The parallel axis theorem lets you shift axes:

$$I=I_{\text{cm}}+M{d}^2$$

Here, $I_{\text{cm}}$ is the moment of inertia about a parallel axis through the center of mass, $M$ is total mass, and $d$ is the distance between the axes.

A common application is a uniform rod rotating about one end instead of its center. You do not need to re-derive $I$ from scratch; you shift it.

Angular momentum and why it is conserved

Angular momentum is the rotational analog of linear momentum. For a rigid body rotating about a fixed axis:

$$L=I\omega$$

The defining relationship for change in angular momentum is:

$${\tau }_{\text{net}}=\frac{dL}{dt}$$

Conservation of angular momentum

If the net external torque on a system is zero, angular momentum is conserved:

$$L_i=L_f$$

This is the principle behind classic AP demonstrations:

• A person on a low-friction turntable spins faster when pulling arms inward.
• A collapsing star spins up as its radius shrinks.

In both cases, $I$ decreases, so $\omega$ increases to keep $I\omega$ constant, assuming negligible external torque.

A crucial detail: angular momentum conservation is about external torque. Internal forces can be large, but they do not change the total angular momentum of the system.

Rotational energy: where the work goes

A rotating object carries kinetic energy:

$$K_{\text{rot}}=\frac{1}{2}I{\omega }^2$$

If something both translates and rotates, total kinetic energy can include both parts:

$$K_{\text{total}}=\frac{1}{2}M{v}^2+\frac{1}{2}I{\omega }^2$$

This shows up in rolling objects and in systems like pulleys with rotational inertia.

Work and power in rotation

Work done by a torque over an angular displacement is:

$$W=\tau \Delta \theta$$

And power in rotational form is:

$$P=\tau \omega$$

These relationships make it possible to connect energy methods to torque methods, and they often simplify multi-step problems.

Rolling without slipping: the capstone connection

Many AP Physics 1 rotational dynamics questions culminate in rolling motion. “Rolling without slipping” means the contact point is instantaneously at rest relative to the surface, producing the key constraint:

$$v_{\text{cm}}=R\omega$$

This is not optional. It is the equation that ties translation to rotation and allows you to use energy conservation cleanly.

A practical insight: for two objects rolling down the same incline, the one with smaller $I$ (for its mass and radius) tends to accelerate more and reach the bottom first, because less of the gravitational potential energy must become rotational kinetic energy.

How to approach AP-style rotational dynamics problems

1. Choose the axis carefully. Picking an axis through a pivot can eliminate unknown torques from forces acting at that pivot.
2. Decide whether to use forces/torques or energy. If nonconservative forces like friction do no net work (as in rolling without slipping with static friction), energy methods can be efficient.
3. Track what is conserved. Use conservation of angular momentum when external torque is negligible. Use conservation of energy when external work is known or can be neglected.
4. Be precise about geometry. Torque depends on the perpendicular lever arm, not just distance. Drawing $r$, $F$, and the angle $\varphi$ prevents sign and sine mistakes.

Rotational dynamics is challenging because it forces you to visualize how forces produce twisting and how mass distribution affects motion. Once the analogs become second nature, the unit becomes less about memorizing formulas and more about choosing the right model: torque and $\alpha$, energy and $\omega$, or conservation laws when the system is isolated.