Angular Displacement: MDCAT Physics notes
Angular Displacement notes for MDCAT: radian and degree conversion, revolutions to radians, s = rθ, right-hand rule, sign convention and vector nature.
What is angular displacement?
When a body moves along a circle, or a rigid body rotates about a fixed axis, the angle swept at the centre by the radius (a line from the axis to the point, perpendicular to the axis) is the angular displacement, $\theta$. The centre (axis) of rotation stays fixed however many revolutions are made.
Units: revolution, degree, radian
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Hence, for arc length $s$ and radius $r$:
$$\theta\ (\text{rad}) = \frac{s}{r} \qquad\text{or}\qquad s = r\theta$$
The radian is a ratio of two lengths, so it relates the arc length to the circumference. One full revolution has arc $2\pi r$, so
$$1\ \text{rev} = 360^\circ = 2\pi\ \text{rad}$$
| Conversion | Value |
|---|---|
| 1 rad | $180^\circ/\pi \approx 57.3^\circ$ |
| $1^\circ$ | $\pi/180\ \text{rad} \approx 0.01745\ \text{rad}$ |
| $\pi$ rad | $180^\circ$ = half revolution |
| $\pi/2$ rad | $90^\circ$ = quarter revolution |
| $2\pi/5$ rad | $72^\circ$ |
| $3\pi$ rad | $3\pi/2\pi = 1.5$ rev |
| 3 rev | $6\pi$ rad |
The Earth rotates through $360^\circ$ ($2\pi \approx 6.28$ rad) in 24 hours, i.e. $15^\circ$ per hour.
Using s = rθ
- Radius 4 m, arc 14 m: $\theta = 14/4 = 3.5\ \text{rad}$.
- Track of diameter 20 m (so $r = 10\ \text{m}$), distance 180 m: $\theta = 18\ \text{rad}$. Always halve the diameter first.
- $r = 1\ \text{m}$, $\theta = 1^\circ$: convert to radians, $s = 1 \times 0.01745 = 0.01745\ \text{m}$.
- For a fixed $\theta$, a smaller radius gives a smaller arc length (linear displacement).
The formula $s = r\theta$ works only with $\theta$ in radians.
Direction and sign
- By convention, anticlockwise rotation is positive and clockwise is negative.
- Direction is given by the right-hand rule: grasp the axis of rotation in the right hand with the fingers curling in the sense of rotation; the thumb points along the angular displacement vector (along the axis).
- Only small angular displacements behave as vectors (they add commutatively). Large rotations do not obey vector addition. For a very small angle, the arc is nearly a straight line, so the angular displacement corresponds to a nearly linear displacement.
Key formulas
- $\theta = s/r$
- $2\pi\ \text{rad} = 360^\circ = 1\ \text{rev}$
- $1\ \text{rad} = 57.3^\circ$; $1^\circ = 0.01745\ \text{rad}$
- Revolutions $= \theta/2\pi$
Common MDCAT traps
- $1\ \text{rad} = 57.3^\circ$, not $1^\circ = 57.3$ rad.
- Given the diameter, use $r = d/2$ in $\theta = s/r$.
- Angular displacement is a vector only for small rotations; it is not "always a scalar" nor "always increasing".
- Direction uses the right-hand rule, not the left-hand or Fleming's rule.
- After one revolution the angle swept is $2\pi$ rad.
Quick revision
- Half revolution = $\pi$ rad.
- Anticlockwise = positive.
- Radian measures arc length relative to radius.
- Earth: $360^\circ$ in 24 h, $30^\circ$ in 2 h.
- $3\pi$ rad = 1.5 revolutions.