12.1 - Radians and angular speed | Cambridge A-Level Physics

Measure angles in radians, describe rotation using angular speed, and connect circular motion to a point's tangential speed. Explain why points on one rigid rotating object share an angular speed while moving at different linear speeds.

Angles measured by arc length

Why is one complete turn $2\pi$ radians, whatever the circle's size? A radian compares two lengths on the same circle. Enlarge the circle and both lengths scale together, so their ratio stays unchanged.

Take an arc of length $s$ on a circle of radius $r$ . The angle $\theta$ at the centre, measured in radians, is $s/r$ . This definition turns angle into a geometrical ratio rather than an arbitrary division of a turn into 360 pieces.

Radian

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

The definition needs both the centre and the arc. A straight chord joining two points on the circle is shorter than the curved arc between them and must not replace it.

[DIAGRAM: asset_name: The radian uses an arc; asset_slug: radian-arc-radius; caption: An arc equal in length to the radius subtends one radian at the centre; description: A circle has two radii separated by exactly one radian,57.3 degrees. The curved arc between them is highlighted and labelled arc length s equals r. The angle at the centre is labelled one radian. A dashed chord is shown separately and labelled not the arc.]
An arc equal in length to the radius subtends one radian at the centre

For a whole circle, $s=2\pi r$ . Hence $\theta=(2\pi r)/r=2\pi$ radians. Half a turn is $\pi$ radians, and a right angle is $\pi/2$ radians. Since arc length and radius both have unit metre, their ratio is dimensionless; the label rad identifies it as an angle.

Arc length and angle conversion

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

Here $s$ and $r$ use the same length unit and $\theta$ is in radians. For conversion,

\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180} \qquad \theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}

Not on the data sheet: learn it.

Worked example 1: an arc and a radius

Question. A point moves through an arc of $0.36\,\mathrm{m}$ on a circle of radius $0.24\,\mathrm{m}$ . Determine its angular displacement in radians and degrees.

Plan. Divide arc length by radius, then convert the radian result.

\theta=\frac{0.36}{0.24}=1.5\,\mathrm{rad} \qquad \theta_{\mathrm{deg}}=1.5\times\frac{180}{\pi}=85.9^\circ

Answer. $1.5\,\mathrm{rad}$ , about $86^\circ$ .

Check. A quarter circumference would be $\pi r/2\approx0.377\,\mathrm{m}$ , slightly longer than the given arc; the angle should be slightly below $90^\circ$ .

For comparison, $150^\circ=150\pi/180=5\pi/6\,\mathrm{rad}$ . Keep the factor of $\pi$ when an exact angle is useful, then use a decimal only when needed.

Watch out

The formula $s=r\theta$ needs $\theta$ in radians. Substituting an angle of 90 as though it were 90 radians gives the wrong arc. Convert degrees first.

Angular speed, period and frequency

Linear speed tells you how quickly distance is covered. Angular speed tells you how quickly the direction from the axis to a rotating point turns. For uniform circular motion, equal angles are swept out in equal times.

Angular speed

Angular speed is the rate of change of angular displacement. Its usual unit is $\mathrm{rad\,s^{-1}}$ .

For a constant angular speed, divide the angle swept out by the time taken. One complete turn sweeps $2\pi$ radians in period $T$ , while frequency $f$ counts turns per second.

Uniform angular speed

\omega=\frac{\Delta\theta}{\Delta t} =\frac{2\pi}{T}=2\pi f

Here $\omega$ is angular speed in $\mathrm{rad\,s^{-1}}$ , $\Delta\theta$ is angular displacement in radians, $\Delta t$ and $T$ are in seconds, and $f$ is in $\mathrm{Hz}$ . For rotation in one sense, the magnitude of $\Delta\theta/\Delta t$ gives the constant or average angular speed over the stated interval.

Not on the data sheet: learn it.

Worked example 2: a rotating platform

Question. A platform completes one turn every $1.6\,\mathrm{s}$ . Determine its angular speed.

Plan. Divide the angle of a whole turn by its period.

\omega=\frac{2\pi}{1.6}\,\mathrm{rad\,s^{-1}}=3.93\,\mathrm{rad\,s^{-1}}

Answer. About $3.9\,\mathrm{rad\,s^{-1}}$ .

Check. In 1.6 seconds it sweeps $3.93\times1.6\approx6.28$ radians, one complete turn.

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12.1 - Radians and angular speed

Angles measured by arc length

Angular speed, period and frequency

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