Unit 5: Torque and Rotational Dynamics

What the CED requires across Unit 5

Unit 5 of AP Physics C: Mechanics is Torque and Rotational Dynamics. The course and exam description weights it at 10 to 15% of the multiple-choice section and suggests about 14 to 20 class periods. That is the middle band for this course: Unit 2 is the heaviest at 20 to 25%, Unit 3 next at 15 to 25%, Unit 4 at 10 to 20%, and Units 1, 5, 6 and 7 all sit at 10 to 15%.

Six topics carry eight learning objectives. Topics 5.3 and 5.4 have two each; the other four have one.

Seven of those eight objectives open with the task verb "describe"; the exception is 5.3.A, which opens with "identify". The CED says the verb describe, used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications", and that within those representations students should be able to describe a physical concept graphically, mathematically, and verbally.

The CED's framing is that Unit 5 reinforces the Unit 2 ideas of force and linear motion by introducing the rotational analogs of torque and rotational motion. It adds that although these topics present more complex scenarios, the tools of analysis remain the same, and that the content and models explored in the first four units set the foundation for Units 5 and 6.

The essential questions printed on the unit opener are deliberately ordinary: why a curveball takes less time to reach the plate than a fastball, why it is easier to balance a bicycle when it is in motion, why long wrenches are more effective, and why it matters where a door handle is placed.

The direction of the torque vector: this is the whole difference

AP Physics 1 has a Unit 5 with the same six topic titles. If you only ever compare one line between the two courses, compare this one.

The AP Physics 1 course and exam description prints a boundary statement under its Topic 5.3. Quoted whole:

"While AP Physics 1 expects students to mathematically manipulate the magnitude of torque using vector conventions, the direction of torque is beyond the scope of the course."

AP Physics C: Mechanics prints no boundary statement at all under its Topic 5.3. What it prints instead is essential knowledge 5.3.B.2, which says the torque exerted on a rigid system about a chosen pivot point by a given force is described by

and then spends three sub-statements unpacking the operation. Statement 5.3.B.2.i gives the magnitude of a cross product of two vectors as $\vec{A} \times \vec{B} = AB \sin\theta$. Statement 5.3.B.2.ii says the direction of the resulting vector is perpendicular to both $\vec{A}$ and $\vec{B}$ and therefore normal to the plane they define. Statement 5.3.B.2.iii says that direction can be qualitatively determined by applying the appropriate right-hand rule.

So the algebra-based course fences the direction of torque out, and the calculus-based course teaches the operation that produces it. The equation sheets agree. The AP Physics C: Mechanics formula sheet prints $\vec{\tau} = \vec{r} \times \vec{F}$, with arrows over both inputs. The AP Physics 1 sheet prints $\tau = r_{\perp}F = rF\sin\theta$, a magnitude with no direction anywhere in it. The C: Mechanics Table of Information also carries a Vectors box that the Physics 1 table does not have at all, and that box prints $\lvert \vec{A} \times \vec{B} \rvert = AB\sin\theta$ and $\vec{A} \cdot \vec{B} = AB\cos\theta$ along with unit-vector notation and vector addition by components.

The direction is not decoration. It is what lets you add two torques about different axes, it is what makes $\vec{L} = \vec{r} \times \vec{p}$ in Unit 6 mean something, and it is why a wheel resists having its plane of rotation changed.

One caution, and it is precise. The unit's other boundary statements do fence off directions, but not the direction of torque. See the next section.

The four boundary statements, and which two topics have none

Unit 5 prints boundary statements under Topics 5.1, 5.2, 5.4 and 5.5. Topics 5.3 and 5.6 print none. Here they are with every clause intact, because the exception clauses are where the real limit lives.

Topics 5.1 and 5.2 carry effectively the same statement, in two paragraphs:

"AP Physics C: Mechanics expects students to be able to mathematically manipulate the magnitudes of angular displacement, angular velocity, and angular acceleration using vector conventions. However, the directions of said vectors will not be assessed on the exam."

"Descriptions of the directions of rotational kinematics quantities for a point or rigid body are limited to clockwise and counterclockwise with respect to a given axis of rotation."

Read that against the previous section and the asymmetry is sharp. The directions of angular velocity and angular acceleration as vectors are outside the exam. The direction of torque is inside it. Clockwise and counterclockwise are all you need for rotational kinematics; a right-hand rule is what you need for a torque.

Topic 5.4 carries the statement that decides how much integration you actually do:

"AP Physics C: Mechanics only expects students to use calculus in the derivations of the rotational inertia of thin rods of uniform or nonuniform density about an arbitrary axis perpendicular to the rod, as well as derivations of the rotational inertia of a thin cylindrical shell, disk, or rigid bodies that can be considered to be made up of coaxial rings or shells about an axis that passes through their centers (e.g., annular rings)."

"Students should have a qualitative understanding of the factors that affect rotational inertia; for example, how rotational inertia is greater when mass is farther from the axis of rotation, which is why a hoop has more rotational inertia than a solid puck of the same mass and radius."

That is a finite list: thin rods about any perpendicular axis, uniform or not; thin cylindrical shells; disks; and bodies built from coaxial rings or shells. A sphere is not on it.

Compare the AP Physics 1 boundary statement for the same topic, which says AP Physics 1 "only expects students to calculate the rotational inertia for systems of five or fewer objects arranged in a two-dimensional configuration", that students "do not need to know the rotational inertia of extended rigid systems, as these will be provided within the exam", and then closes with the same qualitative paragraph, worded with a solid disk rather than a solid puck. Five point masses in the algebra course; a named list of integrals in this one.

Topic 5.5 carries one line: "AP Physics C: Mechanics does not expect students to simultaneously analyze rotation in multiple planes." Note what it does not say. It does not remove the direction of a single torque; it removes having to handle two planes at once.

Rotational inertia stops being a sum

Topic 5.4 hands you both forms and expects you to know which one the object calls for.

Statement 5.4.A.2 gives a single object at perpendicular distance $r$ from the axis as $I = mr^2$. Statement 5.4.A.3 gives a collection as $I_{\text{tot}} = \sum I_i = \sum m_i r_i^2$. Then 5.4.A.4 says that for a solid that can be considered as a collection of differential masses $dm$, the rotational inertia can be calculated using

where $r$ is the perpendicular distance from $dm$ to the axis of rotation. That sentence is the entire content of the sentence "this is a calculus course".

The sum and the integral are not two ways of saying the same thing. The sum can only describe mass that sits at a countable set of distances. The integral describes mass spread continuously, and it is the only route to a rod whose density varies along its length, which the Topic 5.4 boundary statement explicitly puts inside the course. The sheet prints $\lambda = \frac{d}{d\ell}m(\ell)$ for exactly that purpose: a linear mass density you can integrate against.

Statement 5.4.B.1 then says a rigid system's rotational inertia in a given plane is at a minimum when the rotational axis passes through the system's center of mass, and 5.4.B.2 gives the parallel axis theorem as $I' = I_{\text{cm}} + Md^2$. Because $Md^2$ is never negative, those two statements are the same fact stated twice, once in words and once in symbols. The theorem also runs backwards: integrate about a convenient end, subtract $Md^2$, and you have the center-of-mass value without a second integral.

How the six topics build

5.1 Rotational Kinematics defines the three quantities by differentiation. Statement 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. Then 5.1.A.2 gives $\omega = d\theta/dt$ and 5.1.A.3 gives $\alpha = d\omega/dt$. Statement 5.1.A.4.i supplies the three constant-angular-acceleration equations, and 5.1.A.1.i is worth reading twice: a rigid system holds its shape but different points on it move in different directions during rotation, so a rigid system cannot be modeled as an object.

5.2 Connecting Linear and Rotational Motion is the translation layer: $\Delta s = r\Delta\theta$, then $s = r\theta$, $v = r\omega$ and $a_T = r\alpha$ as derived relationships, with 5.2.A.3 stating that all points in a rigid system share the same angular velocity and angular acceleration. Note the subscript on $a_T$. It is the tangential component, not the centripetal one.

5.3 [Torque](/glossary/torque) splits into identifying torques (5.3.A) and describing them (5.3.B). Statement 5.3.A.1 says torque results only from the force component perpendicular to the position vector from the axis to the point of application, and 5.3.A.2 defines the lever arm as the perpendicular distance from the axis to the line of action. Statement 5.3.B.1 introduces force diagrams, which 5.3.B.1.ii distinguishes from free-body diagrams by one feature: force diagrams also depict the location at which each force is exerted relative to the axis of rotation. Then 5.3.B.2 gives the cross product.

5.4 Rotational Inertia is the section above.

5.5 Rotational Equilibrium and Newton's First Law in Rotational Form opens with the statement students most often skip: a system may exhibit rotational equilibrium, meaning constant angular velocity, without being in translational equilibrium, and the other way round. The condition is $\sum \tau_i = 0$.

5.6 Newton's Second Law in Rotational Form gives $\alpha_{\text{sys}} = \frac{\sum \tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}}$, with 5.6.A.2 spelling out that the angular acceleration is in the same direction as the net torque and inversely proportional to the rotational inertia. Statement 5.6.A.3 is the practical warning: to fully describe a rotating rigid system, linear and rotational analyses may need to be performed independently.

Which Unit 5 equations are printed

The CED's Required Equations page states that not all equations in the framework appear on the equation sheet, that many are provided for reference and guidance "or to demonstrate the final results of derivations expected of students on the exam", and that these are denoted Derived Equations. Unit 5 uses that italic label nowhere, although statement 5.2.A.2 does call its three tangential relations derived relationships in the running text. Even so, several of the unit's required equations are not printed.

Three of those rows repay attention.

The single-particle $I = mr^2$ is not printed, and neither is $\sum \tau_i = 0$. Both are one line from something that is printed, which is presumably why. The equilibrium condition is the $\alpha = 0$ case of the rotational second law, and the single particle is the one-term case of the sum.

$\Delta s = r\Delta\theta$ is not printed in that form either. The sheet's version is $\Delta x_{\text{cm}} = r\Delta\theta$, with a center-of-mass subscript, which belongs to rolling in Unit 6. The plain arc-length relation you want in Topic 5.2 is on the sheet as the geometry-box entry $s = r\theta$.

The vectors box does real work here. Because $\lvert\vec{A} \times \vec{B}\rvert = AB\sin\theta$ is printed, the magnitude of a torque never has to be recalled, and because $\vec{C} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$ is printed too, a component-form problem is fully supported. The C: Mechanics sheet also carries a calculus box with the power rule, the sine and cosine derivatives, and the matching integrals. The AP Physics 1 sheet has neither box.

Traps that span more than one topic

A force with a large magnitude can exert zero torque. Statement 5.3.A.1 says torque results only from the force component perpendicular to the position vector. Push a door straight toward its hinge and the torque is zero no matter how hard you push. The cross product says the same thing with $\sin 0 = 0$.

Rotational equilibrium is not translational equilibrium. Statement 5.5.A.1 says a system may be in one without the other. A wheel rolling at constant speed with constant angular velocity is in both. A wheel free-falling while spinning at a constant rate is in rotational equilibrium only.

Rotational equilibrium does not mean at rest. Statement 5.5.A.1.ii defines it as a configuration of torques such that the net torque is zero, and 5.5.A.1.iii states the first law in rotational form: a system has constant angular velocity only if the net torque on it is zero. Constant, not zero.

Choose the axis before you write anything. Statement 5.3.B.2 says "about a chosen pivot point". A torque without a stated axis is not a number. The usual move on a beam problem is to put the axis where an unknown force acts, which drops that force out of the torque equation entirely.

A pulley with mass makes the two tensions different. If the string exerted equal forces on both sides, the net torque on the pulley would be zero and it could not angularly accelerate. The CED's own sample multiple-choice question 10, aligned to learning objective 5.6.A and essential knowledge 5.6.A.1 with skill 3.C, is exactly this: two blocks over a wheel of mass $m$, and the keyed answer is that $T_1 < T_2$ because an unbalanced clockwise torque is needed to accelerate the wheel clockwise.

The distance in a torque is measured from the axis, not from the center of mass. And the distance in $I = \int r^2 dm$ is the perpendicular distance from $dm$ to the axis, which 5.4.A.4 states outright. For a rod rotating about an axis along its own length, every $r$ is zero.

Declare your positive sense of rotation and keep it. Statement 5.1.A.1.ii says one direction of angular displacement about an axis, clockwise or counterclockwise, is typically indicated as mathematically positive, with the other direction becoming mathematically negative. Which one is your choice; changing it mid-problem is a defect.

How Unit 5 is assessed

The AP Physics C: Mechanics exam is 3 hours long. Section I is 42 multiple-choice questions in 85 minutes for 50% of the score. Section II is 4 free-response questions in 95 minutes for the other 50%, always in a fixed order: Mathematical Routines (10 points, suggested 20 to 25 minutes), Translation Between Representations (12 points, 25 to 30 minutes), Experimental Design and Analysis (10 points), and Qualitative/Quantitative Translation (8 points). A four-function, scientific, or graphing calculator is allowed on both sections.

On the multiple-choice section, skill 2.A (derive a symbolic expression) carries 25 to 30%, skill 2.B (calculate or estimate) 20 to 25%, and skills 2.C and 2.D 10 to 15% each. Skill 3.B carries 15 to 25% and 3.C carries 5 to 10%. Practice 1 is not assessed on the multiple-choice section at all. On the free-response section, Practice 1 as a whole carries 20 to 35%, Practice 2 carries 40 to 45%, and Practice 3 carries 30 to 35%.

The unit's Building the Science Practices page flags four skills for Unit 5: 2.A, 2.C, 2.D and 3.B. It says students will be introduced to new but somewhat familiar equations and be expected to derive new expressions from those equations, giving the example of determining the torque exerted on a system if the force exerted is doubled. The unit's Preparing for the AP Exam note then says the analysis of functional relationships is assessed on the fourth free-response question, the Qualitative/Quantitative Translation question, as well as on the multiple-choice section, so students must be able to identify, work with, and predict new values from functional dependencies between variables.

Two of the CED's fifteen sample multiple-choice questions align to Unit 5. Question 9 pairs skill 2.B with objective 5.3.B and essential knowledge 5.3.B.2: a uniform beam of length 0.60 m and mass 4.0 kg pivoted at a wall, held horizontal by a string, with a 1.0 kg block at the far end, asking for the magnitude of the torque about the pivot exerted by the weight of the block. Read the question carefully; the beam's own weight is a distractor here. Question 10 is the pulley problem above.

The sample free-response set's Question 4, a Qualitative/Quantitative Translation question worth 8 points, aligns to objectives 1.3.A, 3.4.B, 5.4.A, 5.6.A, 5.4.B and 6.1.A. It rolls a hollow sphere, a uniform solid sphere and a hoop of identical mass and radius down a ramp and asks students to rank the times, derive the relationship between the time and the rotational inertia, and then predict what happens to a shell filled with liquid that does not rotate with it. Its scoring guidelines award a point for a multistep derivation that uses Newton's second law in rotational form, and add a scoring note that a derivation correctly applying conservation of energy can earn the same points. Two routes, both credited.

Skill 3.A, creating experimental procedures, is listed for Topics 5.4 and 5.6, which is consistent with a course that requires 25 percent of instructional time to be hands-on laboratory work. The unit's five optional sample activities cluster there too: a rotating turntable for 5.1, a suspended walkway and a bike wheel peeling out for 5.3, and for 5.6 a yo-yo whose rotational inertia students find from its downward acceleration plus two rolls of toilet paper dropped one unrolling and one free. The AP Classroom Progress Check for Unit 5 runs about 18 multiple-choice questions and 4 free-response questions, one of each type.

If you are in AP Physics 1, this is not your page

AP Physics C: Mechanics is a calculus-based introductory college-level course, equivalent to the first course in an introductory college sequence in calculus-based physics. Its only stated prerequisite is that students should have taken, or be concurrently taking, calculus.

The algebra-based course has a Unit 5 with the same title, the same six topic titles, and the same 10 to 15% weighting. It is a different unit. Its torque is a signed magnitude, its rotational inertia is a sum over at most five point masses with extended-body values supplied on the exam, and its sheet carries no vectors box and no calculus box.

If that is your course, the page you want is the AP Physics 1 Unit 5 hub. It is written to the AP Physics 1 framework, it stops where that framework stops, and nothing on it is a simplified version of anything here. That page is for AP Physics 1 students; this one is for AP Physics C: Mechanics students. The overlap in titles is a College Board naming decision, not a claim that the material is the same.

For procedure rather than framework, the how to calculate torque guide and the rotational kinematics guide own the step-by-step routines, and the torque calculator checks arithmetic. The AP Physics C: Mechanics course hub lists all seven units.