Topic 5.1: Rotational Kinematics

What Topic 5.1 requires

Topic 5.1 carries one learning objective, 5.1.A: describe the rotation of a system with respect to time using angular displacement, angular velocity, and angular acceleration. Four essential knowledge statements sit under it.

• 5.1.A.1 defines angular displacement as the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis. Its relevant equation is $\Delta\theta = \theta - \theta_0$. Three sub-statements hang off it: 5.1.A.1.i defines a rigid system, 5.1.A.1.ii sets the sign convention, and 5.1.A.1.iii says when a rotating system may be treated as a single object instead.
• 5.1.A.2 defines average angular velocity as the average rate at which angular position changes with respect to time, with the relevant equation $\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}$.
• 5.1.A.3 defines average angular acceleration as the average rate at which the angular velocity changes with respect to time, with the relevant equation $\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}$.
• 5.1.A.4 states that angular displacement, angular velocity, and angular acceleration around one axis are analogous to linear displacement, velocity, and acceleration in one dimension and demonstrate the same mathematical relationships. Sub-statement 5.1.A.4.i lists the three constant-angular-acceleration equations, and 5.1.A.4.ii says that graphs of angular displacement, angular velocity, and angular acceleration as functions of time can be used to find the relationships between those quantities.

One boundary statement applies, and it is short: descriptions of the directions of rotation for a point or object are limited to clockwise and counterclockwise with respect to a given axis of rotation. The identical sentence appears again as the boundary statement for Topic 5.2, so the restriction covers both halves of the rotational kinematics story.

The suggested skills for this topic are 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Those five sit under the three AP Physics 1 science practices: Creating Representations, Mathematical Routines, and Scientific Questioning and Argumentation.

Unit 5 is weighted at 10 to 15 percent of the multiple-choice section, and the CED suggests roughly 15 to 20 class periods for the whole unit. It gives no per-topic day count, so any you see elsewhere is somebody's lesson plan rather than a College Board figure.

Radians are the unit, and radians are dimensionless

Essential knowledge 5.1.A.1 puts the unit inside the definition: angular displacement is measured in radians. That single word decides whether the rest of the topic works.

A radian is defined as the ratio of an arc length to the radius that sweeps it. Both are lengths, so the ratio has no physical dimension. The unit label rad is a reminder of what you measured, not a dimension that has to balance across an equation. That is why radians can appear on one side of an equation and quietly vanish on the other: in $v = r\omega$, meters times radians per second comes out as meters per second, and no conversion factor is needed anywhere.

The numbers you need:

A problem that hands you revolutions, revolutions per minute, or degrees is asking you to convert before you substitute. Do it on the first line of your solution rather than somewhere in the middle.

There is one place students trip over the radian rule in the opposite direction. Not every $\theta$ in rotational physics is an angular displacement. In the torque equation $\tau = r_{\perp}F = rF\sin\theta$, the $\theta$ is the angle between two directions, and it lives inside a sine, so degrees are perfectly fine there and a calculator in degree mode gives the right answer. In $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$, the $\theta$ is an angular position that later gets multiplied by a radius, and only radians make that multiplication correct. Angle inside a trig function: your choice of unit, as long as your calculator agrees. Angle inside a trig function: your choice of unit, as long as your calculator agrees. Angle that will later be multiplied by a radius: radians, and since almost every AP rotation problem ends up doing exactly that, convert on the first line and work in radians throughout.

The three rotational quantities, defined

Angular position $\theta$ is where a marked point on the system sits, measured as an angle from a reference direction about a specified axis. Angular displacement is the change in that position:

Angular displacement is not the same as angle turned through when the system reverses, exactly as displacement is not distance in Topic 1.2. A wheel that turns $+3$ rad and then $-3$ rad has zero angular displacement and has turned through 6 rad of angle.

Average angular velocity is the rate at which angular position changes, from 5.1.A.2:

Its unit is rad/s. Average angular acceleration is the rate at which angular velocity changes, from 5.1.A.3:

Its unit is rad/s squared.

Worth knowing before the exam: none of those three definitional relations is printed on the AP Physics 1 equation sheet. What the sheet does print in its rotational group is the three constant-angular-acceleration equations. The other rotational-kinematics relations on it, $v = r\omega$, $a_T = r\alpha$, and $\Delta x_{\text{cm}} = r\Delta\theta$, belong to Topics 5.2 and 6.5. Check the AP Physics 1 formula sheet yourself and you will see the definitions of $\omega_{\text{avg}}$ and $\alpha_{\text{avg}}$ are not there, in the same way that $v_{\text{avg}} = \Delta x/\Delta t$ is not printed on the linear side. Those two you carry in your head.

One sign note before you go further: a positive $\alpha$ does not mean speeding up, it means the angular velocity is becoming more positive. Speeding up happens when $\omega$ and $\alpha$ share a sign.

The map from linear to rotational, one symbol at a time

Essential knowledge 5.1.A.4 is the reason this topic is fast to learn: the rotational quantities around one axis are analogous to the linear quantities in one dimension and demonstrate the same mathematical relationships. Not similar relationships. The same ones.

Substitute that dictionary into the three constant-acceleration equations from Unit 1 and you get the three the AP Physics 1 sheet prints for rotation, in the same order:

The correspondence runs down to what is missing: the sheet prints three equations on each side and no fourth, so the same selection habit transfers. Pick the equation whose only unknown is the one you want, and if none fits, solve two at once.

The condition transfers too. All three rotational equations assume $\alpha$ is constant over the interval, exactly as their linear parents assume constant $a$. A fan that is switched on and reaches a steady speed has two intervals, not one, and each interval gets its own set of equations. Anything with a smoothly varying angular acceleration is outside what these three can do.

Because the algebra is identical, everything you already know about solving the linear set applies here without translation. If the selection routine itself is what you want to practice, the kinematic equations guide works the linear versions and the rotational kinematics guide works the angular ones step by step. This page stays on what the CED asks you to know rather than repeating those procedures.

Which way is positive, and the boundary that keeps it simple

Essential knowledge 5.1.A.1.ii sets up the sign convention: one direction of angular displacement about an axis of rotation, clockwise or counterclockwise, is typically indicated as mathematically positive, with the other direction becoming mathematically negative.

Read that carefully. It does not say counterclockwise is positive. It says one direction is chosen, and the other one then has to be negative. The choice is yours, it belongs at the top of your solution, and it has to survive to the last line. Counterclockwise positive is the conventional choice, since standard mathematics measures positive angles counterclockwise, and it is a safe default. A problem in which everything turns clockwise is often cleaner the other way, though: call clockwise positive and every number in the problem stays positive.

The topic's boundary statement then caps how far the direction discussion goes: descriptions of the directions of rotation for a point or object are limited to clockwise and counterclockwise with respect to a given axis of rotation. In practice that means the CED does not ask you to describe the direction of a rotating quantity as a vector along the axis. Two words, clockwise or counterclockwise, plus the axis they are measured about, is what the boundary statement treats as a complete direction. Two words, clockwise or counterclockwise, plus the axis they are measured about, is a complete answer.

The phrase "with respect to a given axis of rotation" is doing real work there. Clockwise is not an absolute property of a spinning object; it depends on which side of the axis you view it from. So whenever you write clockwise on a free-response question, make sure the figure or your own sentence has already fixed the axis and the viewing direction.

Rigid systems, and when a system is allowed to be an object

Essential knowledge 5.1.A.1.i gives AP Physics 1 its working definition: a rigid system is one that holds its shape but in which different points on the system move in different directions during rotation, and a rigid system cannot be modeled as an object.

That last clause is the part worth slowing down for, because it connects straight back to Topic 2.1 and the system versus object distinction the course sets up in Unit 2. An object in this course is a thing with no internal structure that matters. A rotating wheel has internal structure that matters a lot: points on opposite sides of the axis move in opposite directions at the same instant, and a rim point covers far more ground than a point near the hub in the same time. Calling it an object throws that away, so the CED does not let you.

Rigid carries weight too. Holding its shape is what keeps the distance from each point to the axis fixed, which is what lets a single $\theta$, $\omega$, and $\alpha$ describe the whole system at once. A stretching spring or a sloshing fluid does not qualify.

Essential knowledge 5.1.A.1.iii gives the escape hatch, and it ships with its own example. If the rotation of a system about an axis may be well described using the motion of the system's center of mass, the system may be treated as a single object. The CED's illustration is Earth: the rotation of Earth about its axis may be considered negligible when considering the revolution of Earth about the center of mass of the Earth and Sun system. Same physical body, two different questions, two different models. Ask about day and night and Earth is a rigid rotating system. Ask about the orbit and Earth is a point mass following its center of mass, which is exactly how Topic 2.9 treats orbiting bodies.

So the modeling question on any Unit 5 problem is not whether the thing is big. It is whether the spin about its own axis matters for what you have been asked.

Angular graphs behave exactly like linear ones

Essential knowledge 5.1.A.4.ii says graphs of angular displacement, angular velocity, and angular acceleration as functions of time can be used to find the relationships between those quantities, and suggested skill 1.B asks you to build quantitative graphs with proper scales and units. The rules are the ones from Topic 1.3, relabelled:

• The **slope of a $\theta$ against $t$ graph** is the angular velocity at that instant.
• The **slope of an $\omega$ against $t$ graph** is the angular acceleration.
• The **area under an $\omega$ against $t$ graph** is the angular displacement $\Delta\theta$.
• The **area under an $\alpha$ against $t$ graph** is the change in angular velocity $\Delta\omega$.
• The **area under a $\theta$ against $t$ graph** means nothing. There is no such quantity in this course.

Shapes worth recognizing on sight: constant angular velocity gives a sloped straight line on $\theta$ against $t$ and a horizontal line on $\omega$ against $t$, while constant non-zero angular acceleration gives a parabola on $\theta$ against $t$, a sloped straight line on $\omega$ against $t$, and a horizontal line on $\alpha$ against $t$.

Linearizing is where skill 1.B and skill 2.A meet, and it shows up on experimental design questions. Suppose you release a spool and measure its angle turned at a set of times. Plotting $\theta$ against $t$ gives a curve you cannot read a number off. Plotting $\theta$ against $t^2$ gives a straight line of slope $\frac{1}{2}\alpha$ for a system starting from rest, so the angular acceleration is twice the slope. Choosing the axes so the theory predicts a straight line, then getting the physical quantity out of the slope, is the standard move on this kind of question.

How Topic 5.1 is tested, and what goes wrong

Unit 5's AP Classroom Progress Check is built from about 18 multiple-choice questions and 4 free-response questions covering Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative or Quantitative Translation. Those are the same four free-response task types the exam itself uses.

The CED's note on Unit 5 says students will be introduced to new but somewhat familiar equations and will be expected to derive new expressions from them, as they have in previous units. Skill 2.D, functional dependence, is listed for this topic, so expect questions of the form "if the angular acceleration is doubled, what happens to the angle turned in the same time?" alongside questions that just want a number.

Five things that cost points:

Mixing degrees and radians. Convert revolutions and degrees to radians on the first line, then keep every angular quantity in radians, because the linear conversions in Topic 5.2 only work that way.

Leaving an answer in radians when the question asked for revolutions. Divide by $2\pi$. The units at the end of the question tell you which one was wanted.

**Using the three equations when $\alpha$ is not constant.** If the question describes a system that speeds up and then holds a steady rate, split the motion at the moment the acceleration changes and treat the two intervals separately.

Flipping the sign convention halfway through. Declare positive once and keep it. A negative angular acceleration paired with a positive angular velocity means slowing down. A negative one paired with a negative angular velocity means speeding up in the negative direction.

Assuming a slower angular velocity means a slower point. It does not, because linear speed also depends on how far the point sits from the axis. That connection is the entire content of Topic 5.2, which turns $\omega$ and $\alpha$ into the speed and tangential acceleration of an actual point on the wheel. From there Unit 5 continues into Topic 5.3 and Topic 5.4, which supply the cause of angular acceleration in the same way force and mass supply the cause of linear acceleration.