Work
Work measures mechanical-energy transfer by a force acting through displacement. The general Physics C model is the line integral W=∫F⃗·dr⃗; constant-force formulas, force–displacement area, conservative-force tests, and the work–energy theorem are all consequences or applications of that model.
Topic sequence · 2 of 5
Learning objectives
Objective 1
Describe positive, negative, and zero work and distinguish conservative from nonconservative work using path dependence.
Objective 2
Calculate work from a dot product, a definite line integral, or signed area under a force-parallel-versus-displacement graph.
Objective 3
Apply and derive the work–energy theorem while choosing an object/system model and tracking the displacement of the force’s point of application correctly.
- Topic 1.1 vectors and dot-product geometry.
- Topic 1.2 derivatives and definite integrals.
- Topic 2.2 free-body diagrams and Topic 2.5 Newton’s second law.
- Topic 3.1 translational kinetic energy.
Definitions first
Representations you must read and create
Representation 1
Force and displacement vectors with the angle θ explicitly marked so the parallel component F cosθ is visible.
Representation 2
F∥ versus displacement graph where signed area represents work in joules.
Representation 3
Energy-transfer diagram distinguishing work done on the selected system from changes in kinetic energy and, later, potential energy.
Core relationships and calculus
Variables: F⃗ force (N); dr⃗ differential displacement of the force point (m)
Use: general work by a variable force along a path
Variables: F constant; d displacement; θ angle between force and displacement
Use: constant parallel-force component
Variables: K translational kinetic energy (J)
Use: work–energy theorem for an object model
Variables: signed area in N·m = J
Use: graphical work for one-dimensional motion
Variables: f_k kinetic-friction magnitude; L path length
Use: mechanical energy dissipated by approximately constant kinetic friction
Work is a line integral because the relevant force component can vary along the path. Parameterize the path or use a coordinate for which F∥ is known, evaluate the definite integral with physical limits, and keep the sign. The work–energy theorem follows by substituting F=m dv/dt and dr=v dt into ∫F·dr, producing ∫m v dv=Δ(½mv²).
For one-dimensional motion, W_net=∫F_net dx. With F_net=m dv/dt and v=dx/dt, write F_net dx=m(dv/dt)(v dt)=m v dv. Integrating from initial to final speed gives W_net=∫m v dv=½mv_f²−½mv_i²=ΔK. The proof shows why the theorem is valid even when the net force varies.
The current Mechanics framework expects analysis of mechanical-energy transfer. Students should recognize that mechanical energy can be dissipated as thermal energy or sound, but the course does not require a full thermodynamic accounting in this unit.
Physical examples
Example 1
A normal force from a smooth track can bend an object’s path while doing zero work when the force remains perpendicular to the instantaneous displacement; the force changes direction of motion without changing speed.
Example 2
For a spring force F_x=−kx, the work from x_i to x_f is ∫(−kx)dx=½k(x_i²−x_f²); the sign follows from the limits and force direction.
Example 3
Kinetic friction over a longer path dissipates more mechanical energy even if the initial and final positions are the same, illustrating path dependence.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Before integrating, identify the system and the displacement of the force’s point of application; that prevents many sign and modeling errors.
Reasoning pattern 2
On an F∥–x graph, work is signed area. Regions below the axis contribute negative work and can cancel positive regions.
Reasoning pattern 3
Use ΔK=W_net for the sum of all works. If only one force’s work is given, do not silently treat it as net work unless the other forces do zero work.
Reasoning pattern 4
For conservative forces, compare paths: if endpoints are identical, the work is identical; a closed path gives zero net conservative work.
Deeper calculus and limiting-case check
In several dimensions, W=∫F⃗·dr⃗ depends on the path unless the force is conservative. A conservative force has zero closed-loop work and can be related to potential energy. Topic 3.3 formalizes that relationship with ΔU=−W_cons and, in one dimension, F_x=−dU/dx.
Model-selection workflow: decide what the problem is before choosing an equation
For Work, the fastest reliable route is not equation hunting. Start by identifying the system boundary, encode it with the energy/work representation, choose the work-energy or conservation equation, and then test the conservative-force and transfer 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 work and dot product as physical quantities first and symbols second.
Choose a representation
Build Force and displacement vectors with the angle θ explicitly marked so the parallel component F cosθ is visible.. If the prompt supplies F∥ versus displacement graph where signed area represents work in joules., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is W = ∫ₐᵇ F⃗·dr⃗. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current Mechanics framework expects analysis of mechanical-energy transfer. Students should recognize that mechanical energy can be dissipated as thermal energy or sound, but the course does not require a full thermodynamic accounting in this unit. 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 and displacement vectors with the angle θ explicitly marked so the parallel component F cosθ is visible. | 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 = ∫ₐᵇ F⃗·dr⃗ | general work by a variable force along a path. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | W = F d cosθ = F∥ d | constant parallel-force component. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Work is a line integral because the relevant force component can vary along the path. Parameterize the path or use a coordinate for which F∥ is known, evaluate the definite integral with physical limits, and keep the sign. The work–energy theorem follows by substituting F=m dv/dt and dr=v dt into ∫F·dr, producing ∫m v dv=Δ(½mv²). | 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 one-dimensional force varies as F_x(x)=6x+2 N from x=1 m to x=3 m. The work is W=∫₁³(6x+2)dx=[3x²+2x]₁³=(27+6)−(3+2)=28 J. If this is the net work on an object, its translational kinetic energy increases by 28 J. The same result is the signed area under the F_x-versus-x graph over the interval. 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 3.2
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 = ∫ₐᵇ F⃗·dr⃗ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Force and displacement vectors with the angle θ explicitly marked so the parallel component F cosθ is visible. and F∥ versus displacement graph where signed area represents work in joules.. 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
Use a force sensor and motion/position sensor while pulling a cart or stretching a spring. Plot the measured force component parallel to displacement versus position. Numerically integrate the force–position data and compare the resulting work with the measured change in kinetic energy, including uncertainty and sensor-zero effects.
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 W=Fd without checking the angle or whether the force component is constant. The general model is the dot-product integral. | Return to the energy/work 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? |
| Treating every force as conservative. Friction and drag are path-dependent and do not define ordinary mechanical potential-energy functions. | Return to the work-energy or conservation 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 displacement of the center of mass automatically when the point of application of an external force moves a different distance. | Return to the energy/work 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 the absolute area under an F–x graph when the question asks for signed work. | Return to the work-energy or conservation 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 3.1 defines the kinetic-energy quantity whose change equals net work. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 2.8 provides a position-dependent spring force that is naturally handled by the work integral. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 2.9 shows a nonconservative velocity-dependent resistive force whose work depends on the actual path and motion. 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 = ∫ₐᵇ F⃗·dr⃗ 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 3.2 mastery ladder
Interactive variable-force work integral
For one-dimensional F∥(x)=ax+b, integrate from x₁ to x₂. The signed result is the area under the force–displacement graph.
Experimental connection
Use a force sensor and motion/position sensor while pulling a cart or stretching a spring. Plot the measured force component parallel to displacement versus position. Numerically integrate the force–position data and compare the resulting work with the measured change in kinetic energy, including uncertainty and sensor-zero effects.
Connections across Physics C
Connection 1
Topic 3.1 defines the kinetic-energy quantity whose change equals net work.
Connection 2
Topic 2.8 provides a position-dependent spring force that is naturally handled by the work integral.
Connection 3
Topic 2.9 shows a nonconservative velocity-dependent resistive force whose work depends on the actual path and motion.
Connection 4
Topic 3.3 will convert conservative-force work into potential-energy functions.
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 W=Fd without checking the angle or whether the force component is constant. The general model is the dot-product integral.
Avoid this
Treating every force as conservative. Friction and drag are path-dependent and do not define ordinary mechanical potential-energy functions.
Avoid this
Using displacement of the center of mass automatically when the point of application of an external force moves a different distance.
Avoid this
Taking the absolute area under an F–x graph when the question asks for signed work.
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
When can I use W=Fd cosθ instead of an integral?
Use it when the force component parallel to the displacement is constant over the displacement. Otherwise use the appropriate definite integral.
Can a force do zero work while still changing motion?
Yes. A force perpendicular to instantaneous displacement can change the direction of velocity without changing speed or kinetic energy.
What is the difference between work by one force and net work?
Work by one force is one contribution. Net work is the sum of work by all forces and equals the object’s change in translational kinetic energy.
Why is friction called nonconservative?
Its work depends on path length and typically converts mechanical energy into thermal or other forms, so no single mechanical potential-energy function describes it.