Curvilinear and circular motion

The idea

On a curved path, velocity and acceleration become vectors. The velocity always points along the path (tangent to it). The acceleration generally doesn’t, because it’s doing two jobs: changing the speed and changing the direction.

Position, velocity and acceleration as vectors

In x–y components: $\mathbf{r}=x\mathbf{i}+y\mathbf{j}\mathbf{v}=\dot{x}\mathbf{i}+\dot{y}\mathbf{j}\mathbf{a}=\ddot{x}\mathbf{i}+\ddot{y}\mathbf{j}$ Magnitudes come from Pythagoras, e.g. $v=\sqrt{{\dot{x}}^2+{\dot{y}}^2}$.

Averages: ${\mathbf{v}}_{avg}=\Delta \mathbf{r}/\Delta t$ uses the straight-line displacement, not the distance along the curve.

Circular motion

For a point moving round a circle of radius $r$ (angle $\theta$ in radians):

$s=r\theta v=r\omega a_t=r\alpha a_n=\frac{v^2}{r}={\omega }^{2}ra=\sqrt{a_t^2+a_n^2}$

• Tangential $a_t$ changes the speed.
• Normal $a_n$, pointing to the centre, changes the direction. Even at constant speed, $a_n\ne 0$.
• Angular versions of the constant-acceleration equations: $\omega ={\omega }_0+\alpha t\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2{\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$

Conversions: 1 rev = $2\pi$ rad · rpm × $\frac{2\pi }{60}$ = rad/s

Worked examples

1. A particle’s position is $\mathbf{r}=2t^2\mathbf{i}+3t\mathbf{j}$ m. Find its speed and acceleration at $t=2$ s.

Answer

$\mathbf{v}=4t\mathbf{i}+3\mathbf{j}=8\mathbf{i}+3\mathbf{j}$ → $v=\sqrt{64+9}=8.54$ m/s $\mathbf{a}=4\mathbf{i}$ → $a=4$ m/s², pointing along x

2. A robot wheel of radius 0.25 m spins at 12 rad/s and is speeding up at 5 rad/s². Find the speed and acceleration of a point on its rim.

Answer

$v=0.25\times 12=3$ m/s $a_t=0.25\times 5=1.25$ m/s², $a_n=12^2\times 0.25=36$ m/s² $a=\sqrt{1.25^2+36^2}=36.0$ m/s². The normal part dominates.

Common mistakes

• Using degrees in $s=r\theta$ or $v=r\omega$. They only work in radians.
• Forgetting $a_n$ when the speed is constant.
• Using distance instead of displacement for average velocity.

Check yourself

1. A point at 0.1 m from an axle turning at 600 rpm. Speed and normal acceleration?

$\omega =600\times 2\pi /60=62.8$ rad/s, $v=6.28$ m/s, $a_n=62.8^2\times 0.1=395$ m/s².