|
potential energy function
|
\[
-\frac{d\,U\left(x\right)}{dx}=F\left(x\right),
\ \ \ U\left(x\right)=-\int F\left(x\right)\,dx
\]
|
|
gravitational potential energy.
$y=$ height above the ground, can let
$C=0$ so $U_g=0$ when $y=0.$
|
\[
\begin{align*}
U_g &=-\int mg\,dy \\
&=mgy+C
\end{align*}
\]
|
|
work done by gravity
|
\[
\begin{align*}
W_g &=-\int_{h_i}^{h_f}mg\,dy\\
&=mgh_i-mgh_f\\
&=U_{g_i}-U_{g_f}\\
&=-\Delta U_g
\end{align*}
\]
|
|
elastic potential energy
|
\[
U_s=\frac{1}{2}kx^2
\]
|
|
work done by a conservative force
|
\[
W_c=-\Delta U_c
\]
|
|
work done by a spring
|
\[
W_s=-\Delta U_s
\]
|
|
conservation of energy for a system.
Holds iff the system is isolated and
contains only conservative forces,
otherwise $\Delta E\neq 0.$
|
\[
\begin{align*}
E_i &=E_f\\
K_i+U_i &=K_f+U_f\\
\Delta K &=-\Delta U\\
\Delta E &=0
\end{align*}
\]
|
|
work-kinetic energy theorem when
conservative and non conservative
forces are present.
$\sum W_c=$ net work done by conservative forces
and
$\sum W_{nc}=$ net work done by non-conservative forces.
|
\[
\begin{align*}
& \sum W_c+\sum W_{nc}=\sum W=\Delta K\\
& -\Delta U+\sum W_{nc}=\Delta K\\
& \sum W_{nc}=\Delta K+\Delta U=\Delta E
\end{align*}
\]
|
|
the
kinetic energy of a system
is
the sum of the kinetic energies of
its particles and of the forces on
them (quick quiz 8.4, p. 250)
|
\[
\begin{align*}
& \sum_{\alpha}{\Delta K_\alpha}=\Delta\sum_{\alpha} K_\alpha=\Delta K\\
& \sum_{\alpha}\left(K_{\alpha_f}-K_{\alpha_i}\right)=\sum_{\alpha} K_{\alpha_f}-\sum_{\alpha} K_{\alpha_i}
\end{align*}
\]
|
|
the
potential energy of a system
is the
sum of the potential energies of its
particles and of the forces on them
(quick quiz 8.4, p. 250)
|
\[
\begin{align*}
& \sum_{\alpha}{\Delta U_\alpha}=\Delta\sum_{\alpha} U_\alpha=\Delta U\\
& \sum_{\alpha}\left(U_{\alpha_f}-U_{\alpha_i}\right)=\sum_{\alpha} U_{\alpha_f}-\sum_{\alpha} U_{\alpha_i}
\end{align*}
\]
|
|
example of work-kinetic energy theorem
when two conservative forces, spring
force and gravitational force, and two
non-conservative forces, applied force
and frictional force, are present.
|
\[
\begin{align*}
& W_g+W_s+W_{\mathrm{app}}+W_k=\sum W=\Delta K\\
& mgy_i-mgy_f+\frac{1}{2}kx_i^2-\frac{1}{2}kx_f^2+W_{\mathrm{app}}-f_kd=\Delta K\\
& W_{\mathrm{app}}-f_kd=\Delta K+mgy_f-mgy_i+\frac{1}{2}kx_f^2-\frac{1}{2}kx_i^2\\
& \sum W_{nc}=\Delta K+\Delta U_g+\Delta U_s=\Delta K+\Delta U=\Delta E
\end{align*}
\]
|
Alex Notes