Simple Harmonic Motion
Motion
Rotational Motion
results from correlated simultaneous oscillations.
x(t)\ =\ X\ \text{cos}(\omega\ t\ + \phi_x)
y(t)\ =\ Y\ \text{cos}(\omega\ t\ + \phi_y)
for
X=Y=A
and
\phi_y=\phi_x-\tfrac{\pi}{2}
x(t)\ =\ A\ \text{cos}(\omega\ t)
y(t)\ =\ A\ \text{sin}(\omega\ t)
\}
x^2+y^2=A^2
RECALL
Position
\vec{r}
relative (to a reference frame)
Velocity
"the instantaneous state of motion" quantifies the rate of change of position
\vec{v}
\vec{v}=\frac{d\vec{r}}{dt}
Acceleration
quantifies changes in the state of motion
\vec{a}
\vec{a}=\frac{d\vec{v}}{dt}=\frac{d^2\vec{r}}{dt^2}
\vec{a}\parallel \vec{v}\iff
change in magnitude of
\vec{v}
change in direction of
\vec{a}\perp\vec{v}\iff
\vec{v}
Description of motion (kinematics)
RECALL
Position
\vec{r}
relative (to a reference frame)
Velocity
"the instantaneous state of motion" quantifies the rate of change of position
\vec{v}
\vec{v}=\frac{d\vec{r}}{dt}
Acceleration
quantifies changes in the state of motion
\vec{a}
\vec{a}=\frac{d\vec{v}}{dt}=\frac{d^2\vec{r}}{dt^2}
\vec{a}\parallel \vec{v}\iff
change in magnitude of
\vec{v}
change in direction of
\vec{a}\perp\vec{v}\iff
\vec{v}
Causes of motion (Dynamics)
| Force | acceleration | |
|---|---|---|
\vec{a}=\frac{\Sigma \vec{F}}{m}
Recall that change in the state of motion is
\vec{a}=\frac{\Sigma \vec{F}}{m}
Recall that to describe motion is to describe the position at every instance in time.