Angular Motion

This topic is contained in pages $12-28$ of the textbook.

</aside>

The radian is the unit used as a measure of angular displacement, $\theta$.

$→ $C=2\pi r$$

→ $C=2\pi r$

The quantity angular displacement is a scalar → $\dfrac{\theta}{2 \pi}=\dfrac{s}{2 \pi r}$ ⇒ $s=r \theta$. The dimensions for $s$ and $r$ cancel, leaving no dimension for $\theta$, only its units radians.

The angular velocity, $\omega$, is defined similarly to linear velocity, as the rate of change of $\theta$ with respect to $t$. This is also true for the angular acceleration (measured in $rad \space s^{-2}$), $\alpha$.

$$ \omega=\dfrac{d\theta}{dt} $$

$$ \alpha=\dfrac{d\omega}{dt}=\dfrac{d^2\theta}{dt^2} $$

The angular motion of an object can be related to its linear motion using the radius of the curved movement of the object.

$$ s=r \theta $$

$$ v=r\omega $$

→ The above can be derived using the linear equation of motion and the first relationship.

$v=\frac{ds}{dt}=\frac{d(r\theta)}{dt}=r\frac{d\theta}{dt}$ → $r$ is a constant, does not change with time, so it is not differentiated

$\frac{d\theta}{dt}=\omega$

$v=r\omega$

$$ a_t=r\alpha $$

The subscript, $t$, denotes that this acceleration is the tangential acceleration (measured in $ms^{-2}$) to the curved motion of the object.

$\omega$ can be related to the time it takes an object to complete a full turn — defined as the period, $T$.

$$ \omega=\dfrac{2\pi}{T} $$

→ $\omega=\dfrac{v}{r}$ → $v=\frac{s}{t}$

$\omega=\dfrac{\frac{s}{t}}{r}=\dfrac{s}{t} \times \dfrac{1}{r}$ → for one full turn, $s=2 \pi r$ and $t=T$

$\omega=\dfrac{2 \pi r}{T} \times \dfrac{1}{r}$

$\omega=\dfrac{2\pi}{T}$

$$ \omega = 2 \pi f $$

→ because $T = \dfrac{1}{f}$, then also $\omega = 2 \pi f$

In general the linear equations of motion mirror the angular equations of motion; when $\alpha = constant$.