Clockwise or counterclockwise motion?

[Bipolarity]

Homework Statement

Th path of the particle P is given by the following equation:

r = <cos(t),sin(t)> defined for all t>0

Is the particle traveling clockwise or counterclockwise as t increases?

Homework Equations

The Attempt at a Solution

I graphed the function, and it is a circle, initially the particle is at the point <1,0> and it moves counterclockwise according to the graph.

But how could I find whether it's traveling clockwise or counterclockwise in general, for any parametric curve?

Might it have anything to do with cross products? This is all assuming the particle's motion lies strictly in the xy plane.

BiP

 

[Mark44]

Homework Statement

Th path of the particle P is given by the following equation:

r = &lt;cos(t),sin(t)&gt; defined for all t>0

Is the particle traveling clockwise or counterclockwise as t increases?

Homework Equations

The Attempt at a Solution

I graphed the function, and it is a circle, initially the particle is at the point <1,0> and it moves counterclockwise according to the graph.

But how could I find whether it's traveling clockwise or counterclockwise in general, for any parametric curve?

Might it have anything to do with cross products? This is all assuming the particle's motion lies strictly in the xy plane.

BiP

Look at
$$ \frac{d \bf{r}}{dt}$$

It will be pointing in the direction of increasing t.

 

[Bipolarity]

Look at
$$ \frac{d \bf{r}}{dt}$$

It will be pointing in the direction of increasing t.

Hey Mark, thanks. I know that v is parallel to the direction of the particle for increasing t. I can calculate it by differentiating each component of r with respect to t.

However, once I know r and v (as functions of t), how can I use them to decide whether the particle travels clockwise, or how can I determine for what values of t the particle travels clockwise?

My friend suggested polar coordinates but I don't know if that works. My idea was cross product of r and v, but is that correct?

BiP

 

[Vargo]

For the example you gave, you just need to know the basic properties of sine and cosine. But in general, if you want to know whether your curve is turning left or right, you need to calculate the acceleration r''(t). If r' cross r'' is positive then that means you are turning left. If r' cross r'' is negative that means it is turning to the right. The cross product comes out to be: x' y'' - y' x''. So you just check whether that is positive (counterclockwise=left turn) or negative (clockwise=right turn)

 

[Dick]

Hey Mark, thanks. I know that v is parallel to the direction of the particle for increasing t. I can calculate it by differentiating each component of r with respect to t.

However, once I know r and v (as functions of t), how can I use them to decide whether the particle travels clockwise, or how can I determine for what values of t the particle travels clockwise?

My friend suggested polar coordinates but I don't know if that works. My idea was cross product of r and v, but is that correct?

BiP

If you are dealing with a plane curve, then sure. The sign of the k component of rxv will tell you which direction you are traveling around the origin.