5.2.1 + 5.2.2(b) - Circular kinematics

Radians and arc length

When an object moves around a circle, its position can be described by the angle swept out at the centre. OCR expects you to use the radian as the natural angle measure for circular motion.

For an angle $\theta$ in radians, radius $r$, and arc length $s$,

This equation is also the definition rearranged: $\theta = s / r$. The ratio $s / r$ has no dimensions because both $s$ and $r$ are lengths, but OCR answers commonly quote angular displacement in rad.

There are $2 \pi$ radians in one full revolution because the full circumference is $2 \pi r$, so

That is why circular-motion formulae use $2 \pi$ so often. If a question gives an angle in degrees, convert it before using $s = r \theta$: $180^\circ = \pi\,\mathrm{rad}$, so $90^\circ = \pi/2\,\mathrm{rad}$.

Worked example: A point on a rotating platform is $0.80\,\mathrm{m}$ from the centre. It moves through an angle of $1.50\,\mathrm{rad}$. The distance travelled along its circular path is

The angle must be in radians for the multiplication to work directly.

[DIAGRAM: asset_slug: 5_02_1_angular_measure_and_circular_kinematics__diagram_01; description: An arc equal in length to the radius subtends one radian.; caption: An arc equal in length to the radius subtends one radian.]

Period and frequency

For repeated circular motion, two timing quantities are especially useful.

Period and frequency are reciprocals:

If a turntable completes one revolution in $0.040\,\mathrm{s}$, its frequency is

If a wheel rotates at $5.0\,\mathrm{Hz}$, it completes five revolutions each second, so the time for one revolution is

A longer period means fewer revolutions each second: doubling $T$ halves $f$. For a point at a fixed radius, each revolution covers the same distance, so halving the number of revolutions per second also halves its speed. The next parts express these timing links using angular velocity.

Be careful with revolutions per minute. Convert to revolutions per second before using hertz. For example, $300\ \text{rev min}^{-1} = 300/60 = 5.00\,\mathrm{Hz}$.