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Physics C: Mechanics Unit 3: Work, Energy and Power

Energy methods solve many problems faster than forces do, especially when forces vary with position. This unit links force and potential energy through calculus and uses energy diagrams to predict motion.

Covers: Work by constant and variable forces, Work–energy theorem, Conservative forces and potential energy, Conservation of mechanical energy, Energy diagrams and equilibrium, Non-conservative work, Power

Study guide

Work measures energy transferred by a force over a displacement. For a constant force, W = F d cos θ, so only the component along the motion counts; a force perpendicular to the motion, like the normal force on a level floor or the tension in a circular orbit, does no work. For a force that varies with position, W = ∫F(x) dx, which is the area under the F–x graph. The spring force F = −kx is the standard example: the work to stretch a spring by x is ½kx², not kx·x, because the force grows from zero.

The work–energy theorem says W_net = ΔK, where K = ½mv². It applies to every force, so it is a general tool. Some forces are conservative: their work depends only on the start and end points, not the path. Gravity and ideal springs are conservative; friction and drag are not. For a conservative force we define potential energy so that W = −ΔU. In one dimension that gives F = −dU/dx, and in more dimensions F = −∇U. Near Earth, U = mgy; for a spring, U = ½kx².

When only conservative forces do work, mechanical energy E = K + U is conserved. When friction or other non-conservative forces act, W_nc = ΔE. A useful habit is to list the energies at two moments and set up E_i + W_nc = E_f. Energy methods give speed easily but not time or direction, so combine them with Newton's second law when you need a force. For example, energy gives the speed at the bottom of a dip or top of a loop, and ΣF = mv²/r then gives the normal force.

An energy diagram plots U(x). A particle with total energy E can only be where U ≤ E, because kinetic energy cannot be negative. Turning points occur where U = E, and the kinetic energy at any point is the vertical gap E − U. Equilibrium points are where dU/dx = 0, so the force is zero. A minimum of U, with d²U/dx² > 0, is stable: a small push produces a restoring force. A maximum is unstable. Reading a U(x) graph or function to find forces, turning points and speeds is a classic test question.

Power is the rate of energy transfer, P = dW/dt. Average power is total work divided by time, and instantaneous power is P = F · v. A motor lifting at constant speed delivers P = mgv. A car at constant speed delivers power equal to the resistive force times speed; since air drag grows as v², cruising power grows roughly as v³. The SI unit is the watt (1 W = 1 J/s), and the kilowatt-hour is a unit of energy, not power.

Key ideas

• Only the force component along the displacement does work.
• Work by a variable force is the area under the F–x graph.
• Net work equals the change in kinetic energy for any forces.
• Conservative forces have potential energies with F = −dU/dx.
• Non-conservative work equals the change in mechanical energy.
• Stable equilibrium sits at a minimum of U(x), and turning points occur where U = E.
• Power is the rate of doing work, and instantaneous power is F · v.

Formulas and rules

Common mistakes

• Using kx² instead of ½kx² for spring energy or spring work.
• Dropping the minus sign in F = −dU/dx.
• Applying conservation of mechanical energy when friction does work.
• Measuring height from inconsistent reference levels at the start and end.
• Confusing average power with instantaneous power.

Before the test, make sure you can

• Calculate work by constant forces at an angle and by variable forces using integrals.
• Apply the work–energy theorem, including friction.
• Find force from a potential energy function, including in two dimensions.
• Solve conservation-of-energy problems with springs, gravity and loops.
• Combine energy and Newton's second law to find normal forces or tensions.
• Locate equilibrium points and classify their stability from U(x).
• Find turning points and maximum speed from an energy diagram.
• Calculate average and instantaneous power.

Words to know

Work

Energy transferred by a force acting through a displacement, W = ∫F · dr.

Kinetic energy

Energy of motion, K = ½mv².

Work–energy theorem

The net work on an object equals its change in kinetic energy.

Conservative force

A force whose work depends only on the start and end points, such as gravity or a spring force.

Non-conservative force

A force whose work depends on the path, such as friction or drag.

Potential energy

Stored energy defined so that the work by a conservative force is −ΔU.

Mechanical energy

The sum of kinetic and potential energy, E = K + U.

Hooke's law

An ideal spring exerts F = −kx, proportional to its stretch or compression.

Energy diagram

A graph of U(x) used to find forces, turning points and allowed regions.

Turning point

A position where U = E, so the kinetic energy is zero and the particle reverses.

Stable equilibrium

A point where dU/dx = 0 and U is a minimum.

Unstable equilibrium

A point where dU/dx = 0 and U is a maximum.

Power

The rate of doing work, P = dW/dt.

Watt

The SI unit of power, one joule per second.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.