Work and energy

Newton's second law tells you the acceleration at each instant. Often you do not need that much detail. You want to know how fast a roller coaster is moving at the bottom of a drop, how high a spring will launch a part, how much power a pump needs, or how far a truck will roll before stopping. Energy methods answer these questions directly, by comparing the start and end of the motion and skipping everything in between.

This chapter defines work, kinetic energy and potential energy, proves the work–energy theorem, and builds the conservation of energy into a practical tool that includes friction losses. It finishes with power and efficiency, the quantities that size motors, pumps and engines.

Work done by a constant force

When a constant force \(\mathbf{F}\) acts on an object that moves through a displacement \(\mathbf{d}\), the work done by the force is the dot product from Linear Algebra:

where \(\theta\) is the angle between the force and the displacement. Work is a scalar, measured in joules: \(1 \text{ J} = 1 \text{ N·m}\). Only the component of force along the motion does work.

• \(\theta < 90^\circ\): positive work. The force helps the motion (a tow rope pulling a car forward).
• \(\theta = 90^\circ\): zero work. The normal force on a car on a level road, and the tension in a string whirling a ball in a circle, do no work.
• \(\theta > 90^\circ\): negative work. The force opposes the motion (friction, brakes).

Holding a heavy box still does no work in this sense, however tiring it is, because there is no displacement. Your muscles use energy internally, but none is transferred to the box.

When several forces act, the net work is the sum of the work done by each, which equals the work done by the net force.

Work done by a varying force

If the force changes with position, split the motion into tiny steps \(dx\), over each of which the force is nearly constant, and add up. That sum becomes an integral:

Graphically, work is the area under the force–displacement curve. For a constant force the area is a rectangle; for a spring, whose force grows linearly with stretch, it is a triangle.

For a spring obeying Hooke's law, the force needed to stretch it by \(x\) is \(F = kx\), where \(k\) is the spring constant in N/m. The work to stretch it from its natural length to \(x\) is

Kinetic energy and the work–energy theorem

Combine Newton's second law with the kinematics equation \(v^2 = v_0^2 + 2a\Delta x\). For a net force \(F\) acting along the motion, \(F\Delta x = ma\Delta x = \tfrac12 mv^2 - \tfrac12 mv_0^2\). This suggests defining the kinetic energy

and gives the work–energy theorem, which holds generally (for varying forces and curved paths too):

The net work done on an object equals its change in kinetic energy. Positive net work speeds it up; negative net work slows it down. Kinetic energy depends on \(v^2\), so doubling speed quadruples it, which is why highway collisions are so much more destructive than parking-lot ones.

Potential energy

Some forces give back the work done against them. Lift a box onto a shelf and the work you did can be recovered when it falls. Compress a spring and it pushes back. Such forces are called conservative, and for them we define a potential energy \(U\): stored energy that depends only on position.

• Gravitational potential energy near the Earth: \(U_g = mgh\), where \(h\) is height above any reference level you choose.
• Elastic potential energy of a spring: \(U_s = \tfrac12 kx^2\), where \(x\) is the stretch or compression.

The work done by a conservative force is the decrease in its potential energy, \(W = -\Delta U\), and it depends only on the start and end points, not the path. Gravity does the same work on a skier who drops 100 m whether she takes the steep run or the gentle one.

Friction and air drag are not conservative. The work they do depends on the path length, and the energy they remove becomes heat that you cannot get back mechanically.

Only changes in potential energy matter, so choose the zero level wherever it is convenient: usually the lowest point in the problem.

Conservation of mechanical energy

If only conservative forces do work (other forces may act, but do no work, like a normal force on a smooth track), the total mechanical energy stays constant:

Energy moves back and forth between kinetic and potential forms, but the sum is fixed.

Notice how much easier this is than Newton's laws. On a curved track, the normal force changes direction and size continuously; solving \(\mathbf{F} = m\mathbf{a}\) would require calculus along the whole path. Energy needs only the two endpoints.

When friction does work

With non-conservative forces such as friction, the mechanical energy changes by the work they do:

Friction makes \(W_{\text{nc}}\) negative: the mechanical energy decreases, and the lost energy appears as heat. Total energy, counting thermal energy, is still conserved. This is the first law of thermodynamics, which you will study in its full form later in the program.

Power

Power is the rate of doing work, or the rate of transferring energy:

The SI unit is the watt: \(1 \text{ W} = 1 \text{ J/s}\). Engineers also meet the kilowatt and, in North American practice, the horsepower (1 hp = 746 W). Energy bills use the kilowatt-hour: 1 kWh = 3.6 MJ, the energy delivered by 1 kW for one hour.

Two machines can do the same work, but the more powerful one does it faster. Raising an elevator 30 m takes the same energy whether it takes 15 s or 60 s; the faster lift needs a motor four times as powerful.

Efficiency

No real machine delivers all the energy it takes in. Some is lost to friction, electrical resistance or turbulence. The efficiency is

always less than 1. Typical values: large electric motors 0.90–0.97, gearboxes 0.95–0.98 per stage, centrifugal pumps 0.60–0.85, car engines 0.25–0.40. Efficiencies multiply through a chain of machines.