Torque and Work
A torque transfers energy when it acts through angular displacement. The general rotational-work relationship is W=∫τ dθ; for constant torque it becomes W=τΔθ, and the signed area under a torque-versus-angle graph gives the work. Net rotational work changes rotational kinetic energy, providing the rotational analogue of the translational work–energy theorem.
Topic sequence · 2 of 6
Learning objectives
Objective 1
Calculate rotational work for constant or variable torque using τ-versus-θ information.
Objective 2
Use the rotational work–energy theorem to connect torque work with changes in angular speed without unnecessary time-domain dynamics.
Objective 3
Interpret the sign of rotational work and relate it to energy transfer, braking, motors, and power.
- Topic 5.3 torque.
- Topic 5.6 Newton’s second law in rotational form.
- Topic 6.1 rotational kinetic energy.
- Topic 3.2 translational work and variable-force integrals.
- Topic 3.5 power.
Definitions first
Representations you must read and create
Representation 1
Torque-versus-angular-position graph with signed area identified as work.
Representation 2
Rotational energy bar chart before and after torque does work.
Representation 3
Wheel/axle diagram showing applied force, lever arm, torque direction, and angular displacement.
Representation 4
Power comparison showing P=τω as the rotational analogue of P=Fv.
Core relationships and calculus
Variables: W J; τ N·m; θ rad
Use: general work by a torque over angular displacement
Variables: constant torque; Δθ in radians
Use: constant-torque special case
Variables: K_rot=½Iω² for rigid fixed-axis rotation
Use: rotational work–energy theorem
Variables: P W; τ N·m; ω rad/s
Use: instantaneous rotational power for the signed one-axis model
Variables: constant I
Use: energy change from initial to final angular speed
Rotational work is a line integral in angular coordinate: W=∫τdθ. For a rigid body with constant I, combine τ=Iα, α=dω/dt, and dθ/dt=ω: τdθ=I(dω/dt)(ωdt)=Iωdω. Integrating gives W_net=∫Iωdω=½Iω_f²−½Iω_i²=ΔK_rot. This is the calculus bridge between rotational dynamics and energy.
For constant I, start from τ_net=Iα. Multiply by dθ and use α=dω/dt plus dθ=ωdt: τ_net dθ=Iωdω. Integrate between initial and final states to obtain ∫τ_netdθ=½Iω_f²−½Iω_i². The left side is net rotational work; the right side is ΔK_rot. For variable τ(θ), the same left-side integral is evaluated directly from the function or graph.
The required Topic 6.2 relationship is work by torque over angular displacement, including signed area under τ versus θ. Use the torque component about the chosen rotational axis and measure θ in radians. Rotational power P=τω and W_net=ΔK_rot are useful derived connections for fixed-axis rigid rotation. If the center of mass also translates, account for that kinetic-energy channel separately; variable-I systems are better handled with angular momentum.
Physical examples
Example 1
A motor applying torque in the same direction as rotation does positive work and raises rotational kinetic energy.
Example 2
A brake torque opposite the angular displacement does negative work and reduces rotational kinetic energy.
Example 3
A variable torque can do the same work as a constant torque if the signed areas under their τ-versus-θ curves are equal.
Example 4
A torque can be large yet do zero work during an interval with zero angular displacement.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
When τ is plotted against θ, find signed area—not slope—to get work.
Reasoning pattern 2
Use energy methods when the problem gives torque and angular displacement but not time; this is often shorter than solving α and then kinematics.
Reasoning pattern 3
Check the sign: torque opposing angular displacement removes rotational kinetic energy.
Reasoning pattern 4
Convert angular displacement to radians before using τΔθ.
Reasoning pattern 5
If both translation and rotation occur, determine whether the requested energy change includes both contributions.
Reasoning pattern 6
When a body both translates and rotates, W_net=ΔK_rot accounts only for the rotational kinetic-energy part of the modeled fixed-axis motion; include translational energy separately when the system model requires it.
Deeper calculus and limiting-case check
Instantaneous rotational power follows directly: P=dW/dt=τ(dθ/dt)=τω. If τ and ω have the same sign, power is positive; opposite signs give negative power. This mirrors P=F·v and provides a quick bridge to motor and braking problems without creating a separate formula-only treatment.
Model-selection workflow: decide what the problem is before choosing an equation
For Torque and Work, the fastest reliable route is not equation hunting. Start by identifying the rotating system and constraint, encode it with the energy/angular-momentum representation, choose the work-energy or angular-impulse equation, and then test the rolling/orbit/isolation 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 rotational work and torque–angle graph as physical quantities first and symbols second.
Choose a representation
Build Torque-versus-angular-position graph with signed area identified as work.. If the prompt supplies Rotational energy bar chart before and after torque does work., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is W = ∫_{θ_i}^{θ_f} τ(θ) dθ. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The required Topic 6.2 relationship is work by torque over angular displacement, including signed area under τ versus θ. Use the torque component about the chosen rotational axis and measure θ in radians. Rotational power P=τω and W_net=ΔK_rot are useful derived connections for fixed-axis rigid rotation. If the center of mass also translates, account for that kinetic-energy channel separately; variable-I systems are better handled with angular momentum. 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 | Torque-versus-angular-position graph with signed area identified as work. | 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 | W = ∫_{θ_i}^{θ_f} τ(θ) dθ | general work by a torque over angular displacement. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | W = τΔθ | constant-torque special case. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Rotational work is a line integral in angular coordinate: W=∫τdθ. For a rigid body with constant I, combine τ=Iα, α=dω/dt, and dθ/dt=ω: τdθ=I(dω/dt)(ωdt)=Iωdω. Integrating gives W_net=∫Iωdω=½Iω_f²−½Iω_i²=ΔK_rot. This is the calculus bridge between rotational dynamics and energy. | 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 wheel with I=2.0 kg·m² starts at ω_i=3.0 rad/s. A variable net torque τ(θ)=4+2θ N·m acts from θ=0 to 2 rad. The work is W=∫₀²(4+2θ)dθ=[4θ+θ²]₀²=12 J. Initial K_rot=½(2)(3²)=9 J, so K_f=21 J. Then 21=½(2)ω_f², giving ω_f=√21≈4.58 rad/s. This reaches the final speed without solving for α(θ) or time. 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 6.2
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 W = ∫_{θ_i}^{θ_f} τ(θ) dθ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Torque-versus-angular-position graph with signed area identified as work. and Rotational energy bar chart before and after torque does work.. 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
Attach a hanging mass or force sensor to a string wrapped around a wheel of known axle radius. Measure torque and angular displacement as the string unwraps. Compare measured work from ∫τdθ (or τΔθ when torque is constant) with the change in ½Iω² obtained from angular-speed measurements. Include uncertainty in radius, force, and angular speed.
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 degrees in W=τΔθ instead of radians. | Return to the energy/angular-momentum 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? |
| Taking absolute area under a τ-versus-θ graph instead of signed area. | Return to the work-energy or angular-impulse 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 one torque instead of net torque when applying W_net=ΔK_rot. | Return to the energy/angular-momentum 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? |
| Assuming nonzero torque always means nonzero work; angular displacement must also occur. | Return to the work-energy or angular-impulse 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? |
| Adding rolling constraints such as v=Rω when the system is only rotating about a fixed axis. | Return to the energy/angular-momentum 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? |
| Integrating the magnitude of a 3D torque vector while θ describes rotation only about one axis; use the signed torque component conjugate to that angular coordinate. | Return to the work-energy or angular-impulse 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? |
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 3.2 is the translational analogue W=∫F·dr. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 3.5 gives the linear power analogue P=F·v. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 5.6 supplies τ=Iα for the work–energy derivation. 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 W = ∫_{θ_i}^{θ_f} τ(θ) dθ 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 6.2 mastery ladder
Interactive torque–angle work integral
For τ(θ)=aθ+b, integrate from θ₁ to θ₂. Enter I and ωᵢ to predict ω_f from W_net=ΔK_rot.
Experimental connection
Attach a hanging mass or force sensor to a string wrapped around a wheel of known axle radius. Measure torque and angular displacement as the string unwraps. Compare measured work from ∫τdθ (or τΔθ when torque is constant) with the change in ½Iω² obtained from angular-speed measurements. Include uncertainty in radius, force, and angular speed.
Connections across Physics C
Connection 1
Topic 3.2 is the translational analogue W=∫F·dr.
Connection 2
Topic 3.5 gives the linear power analogue P=F·v.
Connection 3
Topic 5.6 supplies τ=Iα for the work–energy derivation.
Connection 4
Topic 6.1 supplies K_rot=½Iω².
Connection 5
Topic 6.3 will integrate torque over time instead of angle to obtain angular impulse.
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 degrees in W=τΔθ instead of radians.
Avoid this
Taking absolute area under a τ-versus-θ graph instead of signed area.
Avoid this
Using one torque instead of net torque when applying W_net=ΔK_rot.
Avoid this
Assuming nonzero torque always means nonzero work; angular displacement must also occur.
Avoid this
Adding rolling constraints such as v=Rω when the system is only rotating about a fixed axis.
Avoid this
Integrating the magnitude of a 3D torque vector while θ describes rotation only about one axis; use the signed torque component conjugate to that angular coordinate.
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 rotational version of W=∫F·dr?
It is W=∫τdθ for one-axis rotational work, with θ measured in radians.
What does area under a τ-versus-θ graph represent?
The signed area is the work done by the torque over that angular interval.
When is W=τΔθ valid?
When the torque component about the chosen axis is constant over the angular displacement.
Why can energy methods be faster than τ=Iα?
If torque work and initial energy are known, W_net=ΔK_rot connects directly to final angular speed without solving for time.