Torque Equations

Why This Matters

Torque is the rotational equivalent of force—it's what makes things spin, tip, or stay balanced. On the AP Physics 1 exam, you're being tested on your ability to connect force, lever arm, and rotation into a coherent framework. These equations aren't isolated formulas to memorize; they represent a logical system where each relationship builds on the others. Understanding torque means understanding how a wrench loosens a bolt, why doors have handles far from hinges, and how figure skaters control their spins.

The exam loves to test whether you can identify which torque equation applies in a given scenario—whether you're analyzing a system in rotational equilibrium, calculating angular acceleration from an unbalanced torque, or determining the work and power in rotating systems. You'll see these concepts in both multiple-choice and FRQ contexts, often combined with energy conservation or Newton's laws. Don't just memorize the formulas—know when each equation applies and what physical principle it represents.

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Defining Torque: The Basic Equations

Torque measures how effectively a force causes rotation. The key insight is that both the magnitude of the force and where it's applied relative to the rotation axis matter.

$\tau = rF\sin\theta$ (Scalar Form)

• This is your go-to equation for calculating torque magnitude—it accounts for the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$
• Maximum torque occurs when $\theta = 90°$ because $\sin(90°) = 1$, meaning the force is fully perpendicular to the lever arm
• The sine function isolates the perpendicular component of the force—only the component perpendicular to $\vec{r}$ actually causes rotation

$\tau = Fd_\perp$ (Lever Arm Form)

• The lever arm $d_\perp$ is the perpendicular distance from the axis of rotation to the line of action of the force
• This form is often faster for geometry-heavy problems—especially when forces act at angles and you can identify the perpendicular distance directly
• Equivalent to $rF\sin\theta$ since $d_\perp = r\sin\theta$; choose whichever form matches the information given

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Newton's Second Law for Rotation

Just as $F = ma$ governs linear motion, torque and rotational inertia govern angular acceleration. This is the heart of rotational dynamics.

$\alpha_{sys} = \frac{\sum\tau}{I_{sys}}$ (Rotational Second Law)

• Net torque causes angular acceleration—if $\sum\tau \neq 0$, the object's angular velocity must change
• Rotational inertia $I$ is the resistance to angular acceleration—larger $I$ means the same torque produces less angular acceleration
• This equation is the rotational analog of $a = F_{net}/m$ and appears constantly in problems involving pulleys, rotating disks, and rolling objects

$\tau_{net} = I\alpha$ (Alternative Form)

• Same relationship, different arrangement—use this form when solving for net torque or when the problem gives you $I$ and $\alpha$
• Critical for multi-torque problems where you sum clockwise and counterclockwise torques to find $\tau_{net}$
• Remember sign conventions—typically counterclockwise is positive, but be consistent with whatever convention you choose

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Rotational Equilibrium

When an object isn't accelerating rotationally, the torques must balance. This condition is independent of translational equilibrium—an object can be rotating at constant $\omega$ while accelerating linearly, or vice versa.

$\sum\tau = 0$ (Rotational Equilibrium Condition)

• Zero net torque means constant angular velocity—the object is either stationary or spinning at a steady rate
• Essential for static equilibrium problems involving beams, seesaws, ladders, and hanging signs where nothing rotates
• Choose your pivot point strategically—placing the axis at a point where an unknown force acts eliminates that force from the torque equation

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Work and Power in Rotation

Energy concepts extend naturally to rotational systems. When torque acts through an angular displacement, work is done; when torque acts on a rotating object, power is transferred.

$W = \tau\theta$ (Rotational Work)

• Work equals torque times angular displacement in radians—this is the rotational analog of $W = Fd$
• Angular displacement $\theta$ must be in radians for this equation to work correctly with SI units
• Connects to energy conservation—work done by torque changes the rotational kinetic energy $\frac{1}{2}I\omega^2$

$P = \tau\omega$ (Rotational Power)

• Power is the rate of doing rotational work—torque times angular velocity gives instantaneous power
• Units check: $\text{N·m} \times \text{rad/s} = \text{W}$ (watts), since radians are dimensionless
• Useful for analyzing motors and engines—tells you how much energy per second a rotating system delivers or requires

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Angular Impulse and Momentum Change

Torque applied over time changes angular momentum, just as force applied over time changes linear momentum.

$\Delta L = \tau\Delta t$ (Angular Impulse-Momentum Theorem)

• Angular impulse equals the change in angular momentum—this is the rotational version of $\Delta p = F\Delta t$
• The slope of an $L$ vs. $t$ graph equals net torque—graphical analysis questions love this relationship
• Area under a $\tau$ vs. $t$ graph equals angular impulse—use this for problems with time-varying torque

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Rolling Motion Applications

For objects that roll without slipping, torque from friction creates the angular acceleration that keeps linear and rotational motion synchronized.

$\tau = rF_f$ (Torque from Friction in Rolling)

• Static friction provides the torque that causes angular acceleration in rolling without slipping—it acts at the contact point, a distance $r$ from the center
• Static friction does no work in pure rolling because the contact point has zero instantaneous velocity
• This torque couples linear and angular motion through the constraint $a_{cm} = r\alpha$

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Quick Reference Table

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Self-Check Questions

1. What condition must be true for $\tau = rF\sin\theta$ to give maximum torque, and why does this make physical sense?

2. A disk and a hoop of equal mass and radius experience the same net torque. Which one has greater angular acceleration, and which equation justifies your answer?

3. Compare and contrast the conditions $\sum\tau = 0$ and $\sum F = 0$. Can a system satisfy one without satisfying the other? Give an example.

4. If you're given a graph of torque vs. time, how would you determine the change in angular momentum? What if you're given angular momentum vs. time—how would you find the torque?

5. In rolling without slipping, static friction creates a torque but does no work. Explain how both of these statements can be true simultaneously.