Page Description Equation
kinetic energy \[ K=\frac{1}{2}mv^2 \]
joule. SI unit for work and energy \[ 1\,\mathrm{N\cdot m}=1\,\mathrm{J} \]
200 watt. SI unit for power \[ 1\,\mathrm{W}=1\,\frac{\mathrm{J}}{s} \]
200 horsepower. Unit of power in British engineering system \[ 1\,\mathrm{hp}=746\,\mathrm{W} \]
200 kilowatt hour. Common unit of energy \[ 1\,\mathrm{kWh}=3.60\,\mathrm{MJ} \]
work done by constant force \[ W=\vect{F}\cdot\vect{d}=Fd\cos{\theta} \]
work done by constant force in direction of displacement \[ W=Fd,\text{ where } \theta=0 \]
work done by constant force in direction opposite displacement \[ W=-Fd,\text{ where } \theta=180^\circ \]
work done by force that varies with position. E.g. conservative forces like gravity or a spring. Note, $F_x$ is the component of $\vect{F}$ along the $x\text{-axis}$ and the force at position $x$??? \[ W_x=\int_{x_i}^{x_f}{F_x\,dx},\ \mathrm{where}\ F_x=F\cos{\theta} \]
work-kinetic energy theorem \[ \begin{align*} W_x &=\int_{x_i}^{x_f}{F_x\,dx}\\ &=\int_{x_i}^{x_f}mv\,dv\\ &=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2\\ &=T_f-T_i\\ &=\Delta T \end{align*} \]
work done by friction \[ W_k=-f_kd \]
net work with friction and other forces \[ \sum W=W_{\mathrm{other}}-f_kd \]
work done by a spring \[ \begin{align*} W_s &=\int_{x_i}^{x_f}{-kx\,dx}\\ &=\frac{1}{2}kx_i^2-\frac{1}{2}\ kx_f^2\\ &=K_i-K_f\\ &=-\Delta K \end{align*} \]
work done by a force applied to a spring (pull or push) \[ \begin{align*} W_{\mathrm{app,s}} &=-W_s\\ &=\int_{x_i}^{x_f}kx\,dx\\ &=\frac{1}{2}kx_f^2-\frac{1}{2}kx_i^2\\ &=K_f-K_i\\ &=\Delta K \end{align*} \]
power \[ \begin{align*} \scr{P} &=\frac{dW}{dt}\\ &=\vect{F}\cdot\frac{d\vect{s}}{dt}\\ &=\vect{F}\cdot\vect{v} \end{align*} \]