help with problem circle passing over other circle

[pankun]

Hello,
I am in need of help with the following situation. I appreciate any insight. Imagine a circular laser beam of light of diameter d1 moving at velocity v passing over a circular opening of diameter d2 so the full diameter of d1 passes over the full diameter of d2. The intensity of the laser light moving at v is in W/m2. My goal is to determine how much energy is transferred through the circular opening of diameter d2 as the light of diameter d1 passes over the opening at a velocity of v. I realize this is mainly a physics situation and i´m not worried about the physics portion(intensity, watts, joules etc.) I´m stuck with one circle passing over another circle(assuming they pass along their diameters) at a certain speed to calculate the "amount of area" they overlapped in the time they crossed, to then use that area to do the simple physics calculations.

I hope I have explained the situation clearly. Any help with this would be greatly appreciated.

 

[blamocur]

Hello,
I am in need of help with the following situation. I appreciate any insight. Imagine a circular laser beam of light of diameter d1 moving at velocity v passing over a circular opening of diameter d2 so the full diameter of d1 passes over the full diameter of d2. The intensity of the laser light moving at v is in W/m2. My goal is to determine how much energy is transferred through the circular opening of diameter d2 as the light of diameter d1 passes over the opening at a velocity of v. I realize this is mainly a physics situation and i´m not worried about the physics portion(intensity, watts, joules etc.) I´m stuck with one circle passing over another circle(assuming they pass along their diameters) at a certain speed to calculate the "amount of area" they overlapped in the time they crossed, to then use that area to do the simple physics calculations.

I hope I have explained the situation clearly. Any help with this would be greatly appreciated.

I have several questions:

1. Is [imath]d_1 > d_2[/imath] ?
2. When the beam passes over the hole does the center of the beam passes through the center of the hole.
3. Does the beam move in a straight line?

 

[pankun]

Thank you for responding and I apologize for the lack of clarity leading to your questions.
1. Yes
2. Yes
3. Yes

See attached image.
Thank you very much.

I can describe the approach I'm attempting if you wish. Although I'm extremely interested in your expert suggestion.

 

[Dr.Peterson]

I´m stuck with one circle passing over another circle(assuming they pass along their diameters) at a certain speed to calculate the "amount of area" they overlapped in the time they crossed,

See here:

Circle-Circle Intersection -- from Wolfram MathWorld

Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points. The intersections of two circles determine a line known as the radical line. If three circles mutually intersect in a single point, their point of intersection is the intersection of their...

mathworld.wolfram.com

Equation [14] gives the area you want, in terms of distance d and radii R and r. You can express this in terms of time by taking d as a function of t.

 

[pankun]

Excellent thank you very much. I had not found this in my search and this is very useful. I do appreciate it. I also wonder what your thoughts are on the following approach.
- treating them as half circle functions in the positive y quadrants.
- dividing the area of overlap into two parts, divided by the perpendicular line from the x-axis to the point of intersection.
- taking the area as the sum of the integrals of the two parts of the area.

The calculation could be done until the centers align and then multiply by 4 (x2 for the half circle and x2 for half the time).
If d1 is much larger than d2 so that d2 is fully covered by d1 for some time before the centers allignm would the equation account for this?
Thoughts? What could this look like?
Thank you for your expert thoughts on this.

 

[Dr.Peterson]

You could do it by integration; you'd get an equivalent formula (if you did it right).

If d1 is much larger than d2 so that d2 is fully covered by d1 for some time before the centers allignm would the equation account for this?

In fact, d2 will always be fully covered for some time interval; the formula only applies from "first contact" to "second contact" (to use eclipse language), and from third to fourth. You have to determine those times and account for them separately.

 

[Nine Divines]

This system would be a whole lot easier using focus directrix method.