Physics in Motion

# Angular Variables

Rotational motion occurs when rigid bodies that do not deform rotate. Consider a flat object rotating around an axis that is perpendicular to its plane. A position of a point on the flat object can be described by the coordinates $$r$$ and $$\theta$$, where $$r$$ is the distance from the origin (i.e. the axis) and $$\theta$$ is the angle measured with respect to an arbitrary x-axis. When the object turns an angle of $$\theta$$, the point moves a distance $$s$$ along the arc. The angle $$\theta$$ is defined in radians as the following:

$$\theta = \frac{s}{r}$$

In order to convert from radians to degrees, we use the following ratio:

$$\frac{\theta (radians)}{\theta (degrees)} = \frac{2 \pi}{360^{\circ}}$$

The linear velocity in meters per second of a point as it moves around a circle is called the tangential velocity:

$$v = \frac{ds}{dt} = r \frac{d \theta}{d t}$$

We define the angular velocity $$\omega$$ in radians per second as $$\omega = d \theta / d t$$. Therefore,

$$v \ r \omega$$

If the point is accelerating along its path with tangential acceleration $$a$$, then

$$a = \frac{dv}{dt} = r \frac{d \omega}{dt}$$

We define the angular acceleration $$\alpha$$ in radians per second, as $$\alpha = d \omega / d t = d^2 \theta / d t^2$$. Therefore,

$$a = r \alpha$$