Work and Energy: A Problem-Solving System
Big idea: forces transfer energy through work. Energy methods often solve motion problems without tracking every instant of acceleration.
Learning goals
- calculate work by a constant force and average power;
- use kinetic, gravitational and elastic potential energy;
- apply the work–energy theorem;
- decide when mechanical energy is conserved;
- include friction and efficiency in an energy account.
Work and power
For a constant force and straight displacement, $W=Fs\cos\theta$, where $\theta$ is the angle between force and displacement. Positive work adds kinetic energy; negative work removes it; perpendicular force does zero work. Average power is the rate of energy transfer: $P=W/t$.
Energy stores
Kinetic energy is $K=mv^2/2$. Near a planet's surface, gravitational potential change is $\Delta U_g=mg\Delta h$. An ideal spring stores $U_s=kx^2/2$. The zero level of potential energy is chosen for convenience; only changes affect the physics.
The component $F\cos\theta$ parallel to displacement performs the work. The perpendicular component does not contribute to $W$.
Two conservation tools
The work–energy theorem is always useful: $W_{net}=\Delta K$. If only conservative forces such as gravity and an ideal spring do work, total mechanical energy is conserved: $K_i+U_i=K_f+U_f$. Friction converts mechanical energy into internal energy, so write $K_i+U_i+W_{nc}=K_f+U_f$, where $W_{nc}$ is work by non-conservative forces.