SUMMARY

This unit's results, gathered together for quick revision:

• Work: $W=\vec F\cdot d\vec r = F\,dr\cos\theta$, a scalar, SI unit joule, zero when $F=0$, when $dr=0$, or when $F\perp dr$.
• Constant force: $W=(F\cos\theta)(r_f-r_i)$ = area under the $F\cos\theta$-vs-$r$ graph. Variable force: $W=\int F\cos\theta\,dr$ = area under the (curved) graph.
• Work-kinetic energy theorem: $W=\Delta KE$; and $KE=\tfrac12mv^2=p^2/2m$.
• Potential energy: $U=\int_O^P \vec F_{\text{ext}}\cdot d\vec r$; gravitational $U=mgh$; elastic (spring) $U=\tfrac12kx^2$.
• Conservative forces (gravity, spring, electrostatic): path-independent work, zero work around a closed loop, $F=-dU/dx$. Non-conservative (friction, drag): path-dependent, dissipate mechanical energy.
• Law of conservation of energy: for an isolated system, $E=KE+U=\text{constant}$.
• Vertical circular motion: minimum speed to complete the loop is $\sqrt{gr}$ at the top and $\sqrt{5gr}$ at the bottom; tension is always greater at the bottom by $6mg$.
• Power: $P=W/t$ (average) or $P=dW/dt$ (instantaneous); SI unit watt, $1\text{ hp}=746\text{ W}$; $1\text{ kWh}=3.6\times10^6\text{ J}$ is a unit of energy. Also $P=\vec F\cdot\vec v$. …