Position and Displacement — Question 4

Question 4

An arc and its chord measure different lengths

A marker moves counterclockwise on a circle of radius 4 m centered at O, from angle $30^\circ$ to $150^\circ$, measured from positive x. It follows the shorter arc with no reversal.

1. Find its initial and final Cartesian positions.

2. Find the displacement vector and length.

3. Find the traveled arc length and explain why adding another full turn changes only one of these quantities.

Question 4 - Solution

An arc and its chord measure different lengths

(a) Resolve the two radii.

$$\vec r_i=4(\cos30^\circ,\sin30^\circ)=(2\sqrt3,2)\,\mathrm m,\quad\vec r_f=(-2\sqrt3,2)\,\mathrm m$$ $\vec r_i=(2\sqrt3,2)$ m and $\vec r_f=(-2\sqrt3,2)$ m. Equal vertical coordinates do not mean the positions are the same.

(b) Join the actual endpoints.

$$\boxed{\Delta\vec r=\vec r_f-\vec r_i=(-4\sqrt3,0)\,\mathrm m},\quad|\Delta\vec r|=6.928\,\mathrm m$$ $\Delta\vec r=(-4\sqrt3,0)$ m: due west, magnitude $4\sqrt3\approx6.928$ m. The chord is horizontal even though the particle continually changes its heading along the arc.

(c) Use radians for arc length.

$$s=R\Delta\theta=4\left(\frac{2\pi}{3}\right)=\frac{8\pi}{3}\,\mathrm m,\quad s_{+1}=s+8\pi\,\mathrm m$$ The angular travel is $120^\circ=2\pi/3$ rad, so distance is $R\Delta\theta=8\pi/3\approx8.378$ m. One additional full turn adds $2\pi R=8\pi$ m to distance. It leaves both endpoint positions and therefore the displacement unchanged.