Work
Work is the energy transferred to or from an object via force applied over a displacement, not energy itself.
Work is not energy — it's a measure of energy transfer. In fact, there are only two forms of energy transfer: work and heat. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Whenever a force is applied to an object and causes it to move, work is done on that object.
Key Takeaways
Work (W = Fd cos θ) is the energy transferred to or from an object via force applied over a displacement — not energy itself.
The angle between force and displacement determines how much work is done: maximized at 0°, zero at 90°, negative at 180°.
Gas work (W = PΔV) relates pressure and volume changes to work done by or on a gas — watch for the physics-vs-chemistry sign convention.
Power (P = W/Δt) measures how fast work is performed.
The work-energy theorem (Wnet = ΔK = Kf − Ki) connects net work directly to the change in kinetic energy.
What Is Work?
Work is the dot product of the force vector and the displacement vector, which expands to:
W = Fd cos θ
where:
W is the work done,
F is the magnitude of the applied force,
d is the displacement of the object,
θ is the angle between the force and the direction of displacement.
Force is any interaction that, when unopposed, changes an object's motion — a push or a pull. Displacement is the distance moved in the direction the force is applied. The cos θ term accounts for how the force's direction relates to the direction of displacement.
The Effect of Angle on Work
When force and displacement point in the same direction (θ = 0°), cos θ = 1, and work is maximized. As the angle between them grows, less of the force contributes to displacement, and work decreases.
Angle (θ) | cos θ | Effect on Work |
|---|---|---|
0° (force aligned with displacement) | 1 | Work is maximized |
90° (force perpendicular to displacement) | 0 | No work is done |
180° (force directly opposes displacement) | −1 | Work is negative (energy removed from the object) |
Work Done on Gases
Work also applies to systems involving gases, where it's related to changes in pressure and volume — for example, a gas trapped in a cylinder with a movable piston. The gas can do work on the piston as it expands, or the piston can do work on the gas as it compresses:
W = PΔV
where W is the work done by or on the gas, P is the gas's pressure, and ΔV is the change in the gas's volume.
This relationship can be visualized on a pressure-volume (P-V) diagram, with pressure on the y-axis and volume on the x-axis — the area under the curve represents the work done by or on the system.
Compression (piston moves inward, volume decreases): work is done on the system, typically W < 0.
Expansion (piston moves outward, volume increases): work is done by the system, W > 0.
MCAT Callout — Watch the Sign Convention: The formula above (W = PΔV) is the physics convention: it describes work done by the gas, so an expanding gas does positive work. MCAT General Chemistry material typically uses the opposite chemistry convention, w = −PΔV, which describes work done on the system instead — under that convention, an expanding gas does negative work on itself. Both describe the same physical process; the sign flip comes from which side of the process the formula is tracking. Keep straight which convention a given problem is using.
Power
Power is the rate at which work is performed — it tells you not just how much work is done, but how fast:
P = W/Δt
where P is power, W is the work done, and Δt is the time over which the work is performed. If two engines perform the same amount of work, the one that does it in less time delivers more power.
The Work-Energy Theorem
The work-energy theorem connects work and kinetic energy: the net work done on an object equals the change in its kinetic energy.
Wnet = ΔK = Kf − Ki
where Wnet is the net work done on the object, ΔK is the change in kinetic energy, Kf is final kinetic energy, and Ki is initial kinetic energy.
When work is done on an object, it speeds up or slows down depending on the direction of the force relative to its motion — this change in speed is a change in kinetic energy. The work-energy theorem is especially useful because it lets you find the net work done on an object just from its initial and final speeds, without needing to know the details of the force or the path taken.
Common MCAT Mistakes
Forgetting the cos θ term. Work isn't just force times distance — it's force times distance times the cosine of the angle between them. A force applied perpendicular to motion (θ = 90°) does zero work, even if the object moves a large distance.
Assuming negative work means no motion occurred. Negative work (θ = 180°) means the force removes energy from the object — the object still moves, but the force opposes that motion, as with friction decelerating a sliding block.
Mixing up the physics and chemistry sign conventions for gas work. W = PΔV (physics, work done by the gas) and w = −PΔV (chemistry, work done on the system) describe the same process but flip signs — always check which convention a passage is using before assigning a sign to an expanding or compressing gas.
Treating power and work as the same quantity. Work (W = Fd cos θ) measures total energy transferred; power (P = W/Δt) measures how fast that transfer happens. Two systems can do identical work while having very different power outputs, depending on the time taken.
MCAT-Style Concept Check
Question: A worker pushes a crate with a force of 50 N across a floor, displacing it 4 m. The force is applied at a 60° angle above the direction of the crate's motion. How much work does the worker do on the crate?
A) 50 J
B) 100 J
C) 173 J
D) 200 J
Answer: B
Explanation: Work is W = Fd cos θ = (50 N)(4 m)(cos 60°). Since cos 60° = 0.5, W = (50)(4)(0.5) = 100 J. The angled portion of the force that doesn't align with the displacement doesn't contribute to the work done.
FAQ
What is the formula for work in physics?
Work is calculated as W = Fd cos θ, where F is the magnitude of the applied force, d is the displacement of the object, and θ is the angle between the force and displacement directions.
Why does the angle between force and displacement matter for work?
Only the component of force acting along the direction of displacement contributes to work. At 0°, the force and displacement fully align and work is maximized; at 90°, the force contributes nothing and no work is done; at 180°, the force directly opposes the motion and work is negative.
What's the difference between the physics and chemistry conventions for gas work?
The physics convention, W = PΔV, describes work done by the gas, so an expanding gas does positive work. The chemistry convention, w = −PΔV, describes work done on the system, so an expanding gas does negative work under that convention. Both describe the same physical process — only the sign's reference point differs.
How is power related to work?
Power (P = W/Δt) is the rate at which work is performed — the amount of work divided by the time it takes to do it. Two processes can involve the same amount of work but different power if one takes more time than the other.
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