|
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
|
\[
\begin{align*}
\scr{P} &=\frac{dW}{dt}\\
&=\vect{F}\cdot\frac{d\vect{s}}{dt}\\
&=\vect{F}\cdot\vect{v}
\end{align*}
\]
|
Alex Notes