M2.3 - Work and energy

Energy methods compare two states without requiring the time taken between them. Kinetic energy depends on speed, gravitational potential energy depends on vertical height, and work transfers energy when a force acts through a displacement.

Here $m$ is mass, $v$ is speed, $h$ is height above any consistent datum, and $\phi$ is the angle between a constant force and the displacement. Energy and work are measured in joules. Only changes in potential energy matter, so choose a convenient zero height and preserve the sign of $\Delta h$.

The work-energy principle is equivalently $\Delta K=$ total work done by all forces. Using gravitational potential energy in the ledger avoids also counting gravity as work: choose one representation, not both. On an incline of length $s$ and angle $\theta$, the vertical change is $s\sin\theta$, while a constant resistance of magnitude $R$ dissipates energy $Rs$ in either direction of travel.

A $4$ kg particle moves $5$ m up an incline whose sine is $0.3$. Its initial speed is $8\,\mathrm{m\,s^{-1}}$ and a constant resistance of $6$ N opposes motion. With no driving work,$$\frac12(4)v^2+4g(5\times0.3)=\frac12(4)(8^2)-6(5).$$Thus $2v^2+58.8=98$, so $v=4.43\,\mathrm{m\,s^{-1}}$. The calculation accounts separately for increased potential energy and energy dissipated by resistance.

Power is the rate of doing work, measured in watts, with $1\,\mathrm{kW}=1000\,\mathrm W$. At a particular speed, a parallel driving force is $F=P/v$; therefore constant power produces a smaller driving force at a larger speed. For example, $12$ kW at $20\,\mathrm{m\,s^{-1}}$ corresponds to $600$ N before resistance and weight components are included in $F=ma$.