|
A particle moves with
simple harmonic motion
if its acceleration is proportional to and opposite
its displacement from some origin.
|
\[
\begin{align*}
F &=-kx\\
a &=-\frac{k}{m}x
\end{align*}
\]
|
|
simple harmonic motion, position function
of the particle.
$C$ is the mean value or average value of the function
$x(t).$
(cf. Plane Trigonometry, Rice, Strange, p. 196)
|
\[
\begin{align*}
x &=A\cos{\left(\omega t-\phi\right)}+C\\
&=A\cos{\omega\left(t-\frac{\phi}{\omega}\right)}+C\\
&=A\cos{\omega\left(t-t_0\right)}+C
\end{align*}
\]
|
|
phase
|
\[
\omega t-\phi
\]
|
|
phase constant (phase angle)
|
\[
\phi=\tan^{-1}{\left(-\frac{v_i}{\omega x_i}\right)}
\]
|
|
phase shift,
the value of $t$
for which the argument of the
sinusoidal function is 0. Thus,
the phase shift is $\frac{\phi}{\omega}.$
This value represents some
initial time for the motion.
(cf. Plane Trigonometry, Rice,
Strange, p. 206.)
|
\[
t_0=\frac{\phi}{\omega}
\]
|
|
amplitude (max, peak).
If $\left|A\right|\gt 1$
then the amplitude compared with $\sin{t}$
is increased, if
$\left|A\right|\lt 1$
then it is decreased. The amplitude is
also the maximum or peak of the sinusoid.
|
\[
\left|A\right|=\sqrt{x_i^2+\left(\frac{v_i}{\omega}\right)^2}
\]
|
|
angular frequency (angular speed)
|
\[
\omega=\sqrt{\frac{k}{m}}=\frac{2\pi}{T}=2\pi f
\]
|
|
frequency
|
\[
f=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}
\]
|
|
period
|
\[
T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}
\]
|
|
speed
|
\[
v=\frac{dx}{dt}=-\omega A\sin{\left(\omega t+\phi\right)}
\]
|
|
speed as a function of position
|
\[
v=\pm\omega\sqrt{A^2-x^2}
\]
|
|
acceleration
|
\[
a=\frac{dv}{dt}=-\omega^2A\cos{\left(\omega t+\phi\right)}=-\omega^2x
\]
|
|
initial position
|
\[
x_i=A\cos{\theta}
\]
|
|
initial velocity
|
\[
v_i=-\omega A\sin{\phi}
\]
|
|
max velocity
|
\[
v_{\mathrm{max}}=\omega A
\]
|
|
max acceleration
|
\[
a_{\mathrm{max}}=\omega^2A
\]
|
|
kinetic energy
|
\[
K=\frac{1}{2}m\omega^2A^2\sin^2{\left(\omega t+\phi\right)}
\]
|
|
potential energy
|
\[
U=\frac{1}{2}kA^2\cos^2{\left(\omega t+\phi\right)}
\]
|
|
total mechanical energy
|
\[
E=K+U=\frac{1}{2}kA^2
\]
|
|
simple pendulum angular acceleration
|
\[
\ddot{\theta}=-\frac{g}{L}\theta
\]
|
|
simple pendulum position
|
\[
\theta=A\cos{\left(\omega t+\phi\right)}
\]
|
|
simple pendulum angular frequency
|
\[
\omega=\sqrt{\frac{g}{L}}
\]
|
|
simple pendulum period
|
\[
T=2\pi\sqrt{\frac{L}{g}}
\]
|
Alex Notes