TheIBAxolotl
Work: energy transferred by a force
Work is done when a force moves its point of application. For a constant force \(F\) acting at angle \(\theta\) to a displacement \(s\):
\[W = Fs\cos\theta\]
Only the component of force along the motion does work. A force perpendicular to the motion, such as the tension in a string whirling a ball in a circle, does no work. Work is a scalar measured in joules (\(1\ \text{J} = 1\ \text{N m}\)). For a force that varies, the work done is the area under the force–displacement graph.
Doing work on a system increases its energy; work done by a system decreases it. This is the work–energy principle: the net work done on a body equals its change in kinetic energy.
Forms of mechanical energy
- Kinetic energy: \(E_k = \frac{1}{2}mv^2\), which can also be written \(E_k = \frac{p^2}{2m}\).
- Gravitational potential energy change near the Earth's surface: \(\Delta E_p = mg\Delta h\). Only changes in height matter, so you can choose any convenient zero level.
- Elastic potential energy in a spring obeying Hooke's law: \(E_H = \frac{1}{2}k(\Delta x)^2\), the area of the triangle under the force–extension graph.
Conservation of energy
Energy cannot be created or destroyed, only transferred between stores or forms. If only gravity and elastic forces act (no friction or drag), the total mechanical energy stays constant:
\[E_k + E_p + E_H = \text{constant}\]
When friction or air resistance is present, some energy is transferred to thermal energy of the surroundings. Total energy is still conserved, but mechanical energy is not. Write the energy equation as: initial energy = final energy + energy dissipated.
Power and efficiency
Power is the rate of doing work or transferring energy:
\[P = \frac{\Delta W}{\Delta t}\]
measured in watts (\(1\ \text{W} = 1\ \text{J s}^{-1}\)). For a force moving an object at constant speed \(v\) in the direction of the force, \(P = Fv\). A car cruising at steady speed delivers a driving force equal to the total resistive force, so its engine power is \(P = F_{\text{resistive}}\,v\).
Efficiency compares what you get to what you put in:
\[\eta = \frac{\text{useful energy out}}{\text{total energy in}} = \frac{\text{useful power out}}{\text{total power in}}\]
It has no units and is always less than 1 (or 100 %) in a real device.
Sankey diagrams
A Sankey diagram shows energy flow as arrows whose widths are proportional to the energy (or power) they represent. The input arrow enters on the left; the useful output continues straight on; wasted energy curves away, usually downward. The widths of the outputs must add up to the input width, which is a visual reminder of energy conservation.
Energy density of fuels
Energy density is the energy released per unit volume of fuel, in J m\(^{-3}\). Some sources also quote specific energy, the energy per unit mass in J kg\(^{-1}\). Fuels with high values are prized for transport, where both the mass and the volume carried are limited. Comparing fuels on these measures is a common data-analysis task.
Worked example 1: conservation of energy
A \(0.40\ \text{kg}\) ball is dropped from rest from a height of \(5.0\ \text{m}\). Ignoring air resistance, find its speed just before it lands (\(g = 9.81\ \text{m s}^{-2}\)).
- \(mg\Delta h = \frac{1}{2}mv^2\), so the mass cancels.
- \(v = \sqrt{2g\Delta h} = \sqrt{2 \times 9.81 \times 5.0} = 9.9\ \text{m s}^{-1}\) (2 s.f.)
Worked example 2: efficiency of a motor
An electric motor lifts a \(50\ \text{kg}\) load through \(8.0\ \text{m}\) in \(12\ \text{s}\). The electrical input power is \(450\ \text{W}\).
- Useful energy: \(mg\Delta h = 50 \times 9.81 \times 8.0 = 3.92 \times 10^3\ \text{J}\)
- Useful power: \(\frac{3924}{12} = 327\ \text{W}\)
- Efficiency: \(\frac{327}{450} = 0.73\), or \(73\ \%\)
Common mistakes
- Using the full force instead of its component along the displacement.
- Writing \(E_H = kx\) or \(\frac{1}{2}kx\) instead of \(\frac{1}{2}kx^2\).
- Forgetting to square the speed in \(E_k\), or squaring the mass by accident.
- Giving an efficiency greater than 1, which signals the input and output are swapped.
- Saying "energy is lost" without saying where it goes (usually thermal energy in the surroundings).
Exam tips
- If a question gives heights and asks for speeds, try an energy method before suvat; it avoids directions.
- State the assumption you make, such as "no air resistance" or "all GPE becomes KE".
- In power problems with steady speed, link the driving force to the resistive forces explicitly.
- On a Sankey diagram, check that output widths add up to the input width.
- Keep at least one extra significant figure in intermediate steps.