Work, Energy and Power

Work is the mechanical transfer of energy that happens when a force acts on an object as it moves. For a constant force, the work done is given by:

$$W = F s \cos\theta$$

where $\theta$ is the angle between the force and the displacement. Only the force component along the motion does work: a hiker pulling a sled with a rope at $40^\circ$ to the ground transfers less energy per metre than one pulling horizontally with the same force.

Because work is a scalar, its sign carries the physics. Work is:

• positive when the force component points with the motion
• negative when it opposes it (friction on a sliding box)
• zero when force and displacement are perpendicular, which is why the normal force on a level floor and the tension in a string swinging a ball in a horizontal circle do no work

The definition $W = F s \cos\theta$ is restricted to scenarios where the force is constant over the entire displacement. When dealing with a variable force, such as the restoring force of an ideal spring ($F = k\Delta x$), the calculation requires summing infinitesimally small increments of work, which is equivalent to finding the area under the force-displacement curve.

For an ideal spring this area is a triangle of base $\Delta x$ and height $k\Delta x$, leading directly to the stored elastic energy $E_H = \frac{1}{2}k\Delta x^2$. You must be able to apply $W = F s \cos\theta$ with correct signs and extract work from $F$-$s$ graphs, including curved ones, by estimating area.

Mechanical energy is stored in three forms you must be able to calculate. Kinetic energy belongs to anything moving:

$$E_k = \frac{1}{2}mv^2$$

or equivalently $E_k = \frac{p^2}{2m}$ in terms of momentum $p = mv$, a version that is very handy in collision problems where momentum is the conserved quantity. Because $E_k$ depends on $v^2$, doubling a car's speed quadruples its kinetic energy, which is why stopping distances grow so quickly with speed.

Gravitational potential energy near Earth's surface changes by $\Delta E_p = mg\Delta h$ when an object's height changes by $\Delta h$; only the change is physically meaningful, so you are free to choose any convenient zero level. A $70\ \text{kg}$ climber ascending $200\ \text{m}$ gains $\Delta E_p = 70 \times 9.8 \times 200 \approx 1.4 \times 10^5\ \text{J}$.

Elastic potential energy is stored in a spring or other Hookean material deformed by $\Delta x$, where $k$ is the spring constant in $\text{N m}^{-1}$:

$$E_H = \frac{1}{2}k\Delta x^2$$

The squared dependence again matters: compressing a launcher spring twice as far stores four times the energy. For each store you should be able to compute the energy, track how it changes during a process, and connect those changes to the work done by the relevant force.

In a system where the only forces doing work are conservative, the total mechanical energy $E_k + E_p$ stays constant; energy simply shuffles between stores.

A pendulum is the classic example: at the ends of the swing the energy is entirely gravitational, at the lowest point entirely kinetic, and equating the two lets you predict the maximum speed from the release height without knowing anything about the path in between.

Forces are classified as conservative if the work they do on an object moving between two points is independent of the path taken. Gravity and electrostatic forces are conservative, which allows us to associate a unique potential energy with any position in space.

Non-conservative forces, such as friction and fluid resistance, depend on the path length; the work done by these dissipative forces converts mechanical energy permanently into non-recoverable thermal energy. Consequently, the conservation of total energy always holds, but mechanical energy is only conserved in systems free from non-conservative work.

When friction or air resistance is present, extend the bookkeeping: the mechanical energy lost equals the work done against the dissipative force (taking losses as positive):

$$W_{\text{friction}} = \Delta E_k + \Delta E_p$$

You must be able to set up energy-balance equations for ramps, pendulums, springs and falling objects, with and without friction, and state clearly when mechanical energy conservation is a valid assumption.

Power measures how fast energy is transferred, in watts ($1\ \text{W} = 1\ \text{J s}^{-1}$):

$$P = \frac{\Delta W}{\Delta t}$$

Two motors can do the same work lifting a crate, but the more powerful one does it in less time. For an object driven at speed $v$ by a force $F$ along the motion, substituting $\Delta W = F\Delta s$ gives the very useful form $P = Fv$.

This explains why a car has a maximum speed: at top speed the driving force exactly balances air resistance, so all the engine's useful power goes into pushing against drag, and $v_{\text{max}} = P/F_{\text{drag}}$.

No real device converts all its input into the intended output. Efficiency compares them, quoted as a decimal or percentage:

$$\eta = \frac{E_{\text{useful}}}{E_{\text{total}}} = \frac{P_{\text{useful}}}{P_{\text{total}}}$$

Efficiency is always less than $1$ for real systems because friction and resistive heating divert energy into thermal stores. A motor that is $75\%$ efficient must draw $4/3$ of the useful power it delivers.

You must be able to move fluently between energy and power versions of these equations, combine $P = Fv$ with efficiency in multi-stage problems (engine to wheels, motor to load), and identify where the wasted energy ends up, which is almost always as thermal energy in the surroundings.

Comparing fuels fairly needs two quantities:

• Specific energy is the energy released per unit mass, in $\text{J}\ \text{kg}^{-1}$ (often quoted in $\text{MJ}\ \text{kg}^{-1}$)
• Energy density is the energy released per unit volume, in $\text{J}\ \text{m}^{-3}$

They are linked by the fuel's density: energy density equals specific energy multiplied by density, so a light, bulky fuel can score well on one measure and poorly on the other.

Hydrogen gas is the standard illustration: per kilogram it releases far more energy than petrol, giving it an excellent specific energy, but because the gas is so diffuse its energy density at ordinary pressure is tiny, which is why hydrogen vehicles need heavy high-pressure tanks.

Petrol and other liquid hydrocarbons are popular precisely because they are reasonably good on both measures, while wood and most batteries store much less energy per kilogram and per cubic metre.

Which measure matters depends on the application: aircraft are mass-limited, so specific energy dominates; a fixed storage depot is volume-limited, so energy density matters more.

You must be able to define both quantities, calculate the mass or volume of fuel needed to supply a given energy demand (often combined with an efficiency), and discuss why societies favour fuels with high values of both when evaluating energy resources.