Work selects the force along the displacement
You will be able to: Use a dot product to determine the magnitude and sign of work.
When does a force transfer energy by doing work?
Pull a cart 2 m with a 10 N force. Pulling in the travel direction does 20 J of work. Pulling at 60° does 10 J. The component along the cart’s displacement determines the transfer.
A useful starting point: Kinetic energy depends on the observer →
Words and symbols before equations
- Work W
- Energy transferred by a force acting through displacement, in joules (J).
- Displacement Δr
- Vector from the initial to the final position, in metres.
- Dot product F · Δr
- Multiply matching vector components and add; the result is a scalar.
- Angle θ
- Angle between the force and displacement vectors, not necessarily an angle to the floor.
What this picture assumes
Constant force, displacement along +x, and a translating point object. Force and displacement use separate drawing scales because their units differ; the dashed line is a projection, not another force.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Parallel component 5 N; W = 10 J. At 90° no energy is transferred by this force.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For constant force, W = F · Δr = Fd cos θ, where d is the displacement magnitude. The factor F cos θ is the part of the force along the displacement. Positive work adds energy; negative work transfers it out through that force.
At 90° the work is zero. An ideal stationary track’s normal force can redirect a sliding point object without doing work because it remains perpendicular to the path. Holding a stationary bag also gives zero mechanical work on the bag, even though your body uses energy.
In Cartesian coordinates, W = FₓΔx + FᵧΔy + F_zΔz. This is the same projection rule without needing an angle. Use the displacement of the force’s application point; for our translating point-object model this is also the object’s displacement. Extended deforming systems need additional care.
A worked example, step by step
A constant force F = (3, −2, 4) N acts while its application point moves by Δr = (2, 1, 0) m. Find its work.
- Match the x, y and z components; the directions are fixed throughout the displacement.
- Multiply each pair: 3×2 = 6 J, (−2)×1 = −2 J, and 4×0 = 0 J.
- Add to obtain W = 6 − 2 + 0 = 4 J.
- The z force does no work here because there is no z displacement. The net transfer by this force is positive.
A force can be large and do zero work. Check the displacement and its direction, not effort alone.
Does zero work mean zero force?
Compare with an explanation
No. A perpendicular force or a force with no application-point displacement can do zero work.
Predict. Change one thing. Explain.
Hold F and displacement fixed. Predict work at 0°, 90° and 180°. Change the angle and relate the projection to the signed work graph.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Parallel component 5 N; W = 10 J. At 90° no energy is transferred by this force.
Constant force, displacement along +x, and a translating point object. Force and displacement use separate drawing scales because their units differ; the dashed line is a projection, not another force.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA cart moves by (3, 4, 0) m under a constant force (2, −1, 5) N. Compute the contribution from each axis and explain the sign of the total.
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Compare with the answer and four-point rubric
- 1 point: x contribution is 2×3 = 6 J.
- 1 point: y contribution is (−1)×4 = −4 J.
- 1 point: z contribution is 5×0 = 0 J.
- 1 point: Total W = 2 J: this force transfers net energy into the object.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which angle belongs in Fd cos θ?
The angle between force and displacement.
RECALL 2What is the unit of a force–displacement dot product?
N·m, equal to a joule.
RECALL 3Why can a centripetal force do zero work?
For motion on a fixed circle it is perpendicular to each small displacement.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Work selects the force along the displacement
- Constant force: W = Fd cos θ = FₓΔx + FᵧΔy + F_zΔz.
- θ < 90° gives positive W; θ > 90° gives negative W.
- For changing force or a curved path, use ∫F · dr.
Remember: A force can be large and do zero work. Check the displacement and its direction, not effort alone.
Conditions: Constant force, displacement along +x, and a translating point object. Force and displacement use separate drawing scales because their units differ; the dashed line is a projection, not another force.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.2, objectives 3.2.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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