## An ant on a rod

A slim rod OA of length R rotates with angular velocity ω in a clockwise direction in a plane around point O. There is an ant crawling along the rod from the point O to the point A with constant speed $$\vec{v}(t)$$ (measured with regard to the rod). Determine the time dependent location of the ant in the laboratory reference frame. Assume its position at time t = 0 s was at the centre of the rod.

• #### Hint 1: A picture

Let’s put point O of the rod at the origin of the coordinate system and let the rod begin at the x-axis at time t = 0 s. Draw a picture including the starting positions of the rod and the ant and also their positions at some random time t. Mark the position vector $$\vec r$$ of the ant at time t.

• #### Hint 2: Position vector of an ant

What is the distance the ant crawled in time t and what is the length of the position vector of the ant at time t? What is the angle the rod has turned in time t?

Using the angle α determine the x and y components of the position vector $$\vec{r}\left(t\right)$$.

• #### Complete solution

Picture:

Let’s split the motion into two parts; a uniform linear motion with velocity v and a rotary motion with angular velocity ω:

The ant has crawled a distance along the rod of vt.

The length of the position vector at time t is therefore (since the ant began at the centre of the rod):

$r(t)=vt+\frac{R}{2}\tag{1}$

The rod has reached an angle:

$\alpha=\omega{t}$

The length of the position vector changes with time according to equation (1):

$r(t)=vt+\frac{R}{2}$

The x component of the position vector therefore equals

$x(t)=r(t)\cos\alpha=\left(vt+\frac{R}{2}\right)\cos\omega{t}$

The y component of the position vector equals

$y(t)=-r(t)\sin\alpha=-\left(vt+\frac{R}{2}\right)\sin\omega{t}$

Expressing the position vector as the vector sum of its components:

$\vec{r}(t)= x(t)\vec{i} + y(t)\vec{j}$

where$$\vec{i}$$, $$\vec{j}$$ are unit vectors in directions x and y

$\vec{r}(t)= \left(vt+\frac{R}{2}\right)\cos\left(\omega{t}\right)\vec{i}-\left(vt+\frac{R}{2}\right)\sin\left(\omega{t}\right) \vec{j}$

$\vec{r}(t)= \left(vt+\frac{R}{2}\right)\cos\left(\omega{t}\right)\vec{i}-\left(vt+\frac{R}{2}\right)\sin\left(\omega{t}\right) \vec{j}$
where $$\vec{i}$$, $$\vec{j}$$  are unit vectors in the corresponding directions x and y.