Torque
Torque measures the rotational effect of a force about a chosen axis. In vector form τ⃗=r⃗×F⃗, so only the component of force perpendicular to the position vector contributes. The same torque magnitude can be written rF sinφ or F times the perpendicular lever arm. Physics C requires both magnitude/sign reasoning and the cross-product/right-hand-rule interpretation.
Topic sequence · 3 of 6
Learning objectives
Objective 1
Identify which forces exert torque about a specified axis and determine the correct lever arm or perpendicular force component.
Objective 2
Calculate torque using τ⃗=r⃗×F⃗, rF sinφ, lever-arm, and component representations with consistent direction/sign conventions.
Objective 3
Create force/torque diagrams that show both force vectors and their points of application, then justify how geometry changes rotational effect.
- Topic 1.1 vectors and components.
- Topic 2.2 force diagrams.
- Topic 5.1 rotational sign conventions.
- Topic 5.2 radius-based linear–rotational geometry.
Definitions first
Representations you must read and create
Representation 1
Force diagram of a rigid body showing the pivot, force vectors, and each point of application.
Representation 2
Position vector r⃗ and force F⃗ with angle φ labeled so rF sinφ can be read directly.
Representation 3
Lever-arm construction showing the perpendicular distance from pivot to line of action.
Representation 4
Signed torque table listing each force, lever arm, clockwise/counterclockwise sense, and τ value.
Core relationships and calculus
Variables: τ⃗ torque (N·m); r⃗ position vector (m); F⃗ force (N)
Use: general torque vector about a chosen axis/origin
Variables: φ is the angle between r⃗ and F⃗
Use: torque magnitude from force angle
Variables: ℓ perpendicular lever arm (m)
Use: equivalent lever-arm calculation
Variables: planar x–y problem; z is perpendicular to plane
Use: component form of the cross product for two-dimensional rotation
Variables: signed torques about the same chosen axis
Use: net torque from multiple forces
Torque is built from a cross product rather than a time derivative or integral, but its vector structure is central to later calculus. Topic 5.6 links net torque to angular acceleration through Στ=Iα, while Unit 6 integrates torque over angle to obtain rotational work and over time to obtain angular impulse. Mastering r⃗×F⃗ here prevents sign and geometry errors later.
Decompose F⃗ into components parallel and perpendicular to r⃗. The parallel component points along the radius and has no rotational leverage. The perpendicular component has magnitude F sinφ, so |τ|=rF⊥=rF sinφ. Geometrically, ℓ=r sinφ is the perpendicular lever arm, giving the equivalent form |τ|=Fℓ. The cross product packages both magnitude and the perpendicular direction into τ⃗=r⃗×F⃗.
Topic 5.3 focuses on identifying and calculating torques about a chosen axis, including the cross product and right-hand-rule direction. Conditions for zero net torque are developed in Topic 5.5, while the quantitative relation between nonzero net torque and angular acceleration belongs to Topic 5.6.
Physical examples
Example 1
A force directed exactly through the pivot has zero lever arm and therefore zero torque even if the force itself is large.
Example 2
The same force applied farther from a hinge produces a larger torque because the lever arm increases.
Example 3
For fixed r and F, torque is largest when the force is perpendicular to r⃗ and zero when the force is parallel or antiparallel to r⃗.
Example 4
In a planar beam problem, choosing counterclockwise torque as positive converts the vector direction into a consistent signed scalar equation for that one axis.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Mark the axis first. A torque value is meaningless unless the axis/pivot about which it is calculated is clear.
Reasoning pattern 2
Use either rF sinφ or Fℓ, not both simultaneously as extra factors.
Reasoning pattern 3
A force can be nonzero yet exert zero torque about a particular pivot if its line of action passes through that pivot.
Reasoning pattern 4
For planar problems, choose a positive rotational sense once and keep it throughout the torque sum.
Reasoning pattern 5
For vector questions, use the right-hand rule on r⃗×F⃗ rather than on F⃗×r⃗; reversing the order reverses the torque direction.
Deeper calculus and limiting-case check
The planar component expression τ_z=xF_y−yF_x follows directly from the determinant form of r⃗×F⃗. It is a useful algebraic check when a force and point of application are given by components. Limiting cases also verify the model: φ→0 gives τ→0; φ→90° gives the maximum magnitude rF.
Model-selection workflow: decide what the problem is before choosing an equation
For Torque, the fastest reliable route is not equation hunting. Start by identifying the axis and rigid-body model, encode it with the torque/rotation representation, choose the rotational Newton-law equation, and then test the axis choice and rigid-body assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat torque and position vector as physical quantities first and symbols second.
Choose a representation
Build Force diagram of a rigid body showing the pivot, force vectors, and each point of application.. If the prompt supplies Position vector r⃗ and force F⃗ with angle φ labeled so rF sinφ can be read directly., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is τ⃗ = r⃗ × F⃗. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Topic 5.3 focuses on identifying and calculating torques about a chosen axis, including the cross product and right-hand-rule direction. Conditions for zero net torque are developed in Topic 5.5, while the quantitative relation between nonzero net torque and angular acceleration belongs to Topic 5.6. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Force diagram of a rigid body showing the pivot, force vectors, and each point of application. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | τ⃗ = r⃗ × F⃗ | general torque vector about a chosen axis/origin. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | |τ⃗| = rF sinφ | torque magnitude from force angle. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Torque is built from a cross product rather than a time derivative or integral, but its vector structure is central to later calculus. Topic 5.6 links net torque to angular acceleration through Στ=Iα, while Unit 6 integrates torque over angle to obtain rotational work and over time to obtain angular impulse. Mastering r⃗×F⃗ here prevents sign and geometry errors later. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A 30 N force is applied 0.40 m from a pivot at 35° to the radial direction. The torque magnitude is τ=rF sin35°=(0.40)(30)sin35°≈6.88 N·m. The lever arm is ℓ=r sin35°≈0.229 m, giving the same τ=Fℓ. If the force tends to rotate the object clockwise and counterclockwise is chosen positive, the signed torque is −6.88 N·m. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 5.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship τ⃗ = r⃗ × F⃗ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Force diagram of a rigid body showing the pivot, force vectors, and each point of application. and Position vector r⃗ and force F⃗ with angle φ labeled so rF sinφ can be read directly.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Mount a meterstick or lightweight beam on a low-friction pivot. Apply known forces at several radii and angles using spring scales. For each trial, calculate rF sinφ and Fℓ, compare the two torque estimates, and identify configurations that produce the same torque despite different r and F.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using the full force magnitude when the force is not perpendicular to r⃗; only F⊥ contributes. | Return to the torque/rotation representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Measuring r to the wrong point. r⃗ ends at the actual point where the force is applied. | Return to the rotational Newton-law equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using the distance from the pivot to the force vector instead of the perpendicular distance to the force’s line of action when using τ=Fℓ. | Return to the torque/rotation representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Mixing clockwise/counterclockwise signs after starting a planar scalar torque sum. | Return to the rotational Newton-law equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating N·m as joules merely because the base units match; torque and energy are different physical quantities. | Return to the torque/rotation representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.2 supplies the force-diagram discipline used in torque diagrams. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 5.4 explains how mass distribution determines the rotational response to a given torque. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 5.5 sets the net torque to zero for rotational equilibrium. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write τ⃗ = r⃗ × F⃗ and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 5.3 mastery ladder
Interactive torque geometry check
Calculate torque magnitude and lever arm from r, F, and the angle between r⃗ and F⃗.
Experimental connection
Mount a meterstick or lightweight beam on a low-friction pivot. Apply known forces at several radii and angles using spring scales. For each trial, calculate rF sinφ and Fℓ, compare the two torque estimates, and identify configurations that produce the same torque despite different r and F.
Connections across Physics C
Connection 1
Topic 2.2 supplies the force-diagram discipline used in torque diagrams.
Connection 2
Topic 5.4 explains how mass distribution determines the rotational response to a given torque.
Connection 3
Topic 5.5 sets the net torque to zero for rotational equilibrium.
Connection 4
Topic 5.6 connects nonzero net torque to angular acceleration.
Connection 5
Unit 6 later uses torque in rotational work and angular-impulse relationships.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using the full force magnitude when the force is not perpendicular to r⃗; only F⊥ contributes.
Avoid this
Measuring r to the wrong point. r⃗ ends at the actual point where the force is applied.
Avoid this
Using the distance from the pivot to the force vector instead of the perpendicular distance to the force’s line of action when using τ=Fℓ.
Avoid this
Mixing clockwise/counterclockwise signs after starting a planar scalar torque sum.
Avoid this
Treating N·m as joules merely because the base units match; torque and energy are different physical quantities.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is the difference between r and the lever arm?
r is the distance from pivot to the force’s point of application; the lever arm is the perpendicular distance from the pivot to the force’s line of action.
When is torque maximum?
For fixed r and F, the magnitude rF sinφ is maximum when the force is perpendicular to the position vector.
Can a large force produce zero torque?
Yes. If its line of action passes through the chosen pivot, the lever arm is zero.
Why does torque use N·m instead of joules?
The units are dimensionally identical, but torque is a rotational vector effect of force whereas energy is a scalar. The quantities should not be treated as interchangeable.