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,

$$ a = r \alpha $$