## An ant on a rod

### Task number: 384

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

\[r(t)=vt+\frac{R}{2}\tag{1}\]*t*is therefore (since the ant began at the centre of the rod):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

\[x(t)=r(t)\cos\alpha=\left(vt+\frac{R}{2}\right)\cos\omega{t}\]*x*component of the position vector therefore equalsThe

\[y(t)=-r(t)\sin\alpha=-\left(vt+\frac{R}{2}\right)\sin\omega{t}\]*y*component of the position vector equalsExpressing 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

\[\vec{r}(t)= \left(vt+\frac{R}{2}\right)\cos\left(\omega{t}\right)\vec{i}-\left(vt+\frac{R}{2}\right)\sin\left(\omega{t}\right) \vec{j}\]*x*and*y*#### Answer

The time dependent position vector of the ant is:

where \(\vec{i}\), \(\vec{j}\) are unit vectors in the corresponding directions

*x*and*y*.