5.1 Rotational Kinematics

Before the Fall 2024 course update, this topic was part of one big Rotation unit (old Unit 5). That unit is now split in two: Unit 5 covers spinning motion, torque and rotational inertia, and Unit 6 covers rotational energy, angular momentum and rolling. Older videos and practice labeled "Unit 5: Rotation" still fit, but they may mix in Unit 6 ideas.

Angular position θ tells you how far an object has turned from a reference line. In physics you measure it in radians. One radian is the angle where the arc length equals the radius, so $\theta = \dfrac{s}{r}$. A full turn is 2π rad, which is 360°.

Angular displacement Δθ is the change in angular position. One direction of turning (usually counterclockwise) is called positive and the other negative. The problem may pick for you; if not, pick one and stick with it.

A rigid system is an object that keeps its shape while it turns, like a wheel or a door. Different points on it move in different directions at the same moment, so you can't treat it as a single point. If the spin doesn't matter for the question, you can still model it as one object. For example, you can ignore Earth's daily spin when you study its yearly orbit around the Sun.

Angular velocity is how fast the angular position changes: $\omega = \dfrac{d\theta}{dt}$, in rad/s. Angular acceleration is how fast the angular velocity changes: $\alpha = \dfrac{d\omega}{dt}$, in rad/s².

Signs work just like in one-dimensional motion. If ω and α have the same sign, the object is spinning faster. If they have opposite signs, it's slowing down. When ω passes through zero, the object stops for an instant and reverses its direction of spin.

Problems often give speeds in revolutions per minute (rpm). Convert before you use any formula: multiply by 2π rad per revolution and divide by 60 s per minute.

Everything you learned about x, v and a graphs works here with θ, ω and α. The slope of a θ–t graph is ω, and the slope of an ω–t graph is α. The area under an ω–t graph is Δθ, and the area under an α–t graph is Δω.

When α changes with time, integrate: $\omega(t) = \omega_0 + \displaystyle\int_0^t \alpha\,dt$ and $\theta(t) = \theta_0 + \displaystyle\int_0^t \omega\,dt$. Don't forget the starting values ω₀ and θ₀.

When α is constant, you can use the rotational versions of the Unit 1 kinematic equations. Each one is the linear equation with x, v and a swapped for θ, ω and α:

$$\omega = \omega_0 + \alpha t \qquad \theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2 \qquad \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$$

These only work for constant α. If α depends on time, go back to integrals.

As vectors, ω and α point along the axis of rotation, set by a right-hand rule. The exam won't ask for those directions. You'll only describe rotation as clockwise or counterclockwise about a given axis, and you'll work with the signed sizes of θ, ω and α.