Care to help me with my physics homework?

[zahc]

There seems to be a few nerds and engineering types here.

A camera at the red dot is following a car that is going around the track at a known speed V. a, b are known. What is the angular velocity the camera must turn to keep the car framed, based on the angle and V?



I hope you can see the tiny diagram.

Obviously the angular velocity (W)=(the component of V perpendicular to the R)/r, but I can't find a way to express it. It was easy for the straight part. I got that with simple geometry but I can tell the curved part requires a vector approach.

[TarpleyG]

Well, all I know for sure is that I can do it in real life because I have before.  Sorry that doesn't help you any.  I made a A in physics in HS too.  So much for retained knowledge.

Greg

[zahc]

If I could simply get a unit vector perpendicular to R, and express V in terms of the knowns, I could simply do the dot product.

I can easily get the angular velocity at the center of the circle so if I could somehow relate the two angles, one as a function of the other, it would be done. I've just been chasing the math around and getting nowhere.

[One of Many]

You are not dealing with a constant angular velocity at the camera.  The distance between the camera and the vehicle is not changing at a constant rate, based on the angle shown at the camera.  The vehicle at a constant tangential velocity on the circular curve, has an angular acceleration relative to the radius of the circular curve.  Even on the straight section of the track, the angular velocity at the camera is not a constant; it is changing as the vehicle distance from the camera varies.  The maximum angular velocity at the camera, occurs as the vehicle is at the shortest distance from the camera; the minimum angular velocity occurs at the longest distance between the two.  The rate of change of the angular acceleration of the camera is also changing when the vehicle enters the curve.

[zahc]

Sure, I'm to find the angular velocity of the camera as a function of theta. I mapped it out for all the straight section (simple) but the curve is going to require a vector approach. I'm stuck.

[One of Many]

Try to determine the distance between camera and vehicle, based on the angle Alpha of the vehicle relative to the center of the circular section.  There are two angular velocities involved, and the circular angle Alpha dominates in that portion of the path.  Find the distance r where, (a < r < a+b), and the height Y of the vehicle on the circle is the sine of the angle Alpha multiplied by b, the horizontal addendum Z to distance a is the cosine of the angle Alpha multiplied by b, and Theta = inverse tangent (Y/a+Z) with Y = b(sine Alpha) and Z = b(cosine Alpha). Theta = inverse tangent {b(sine Alpha) / [a + b(cosine Alpha)]}.

With a constant tangential velocity V, the values of Y and Z are accelerating as the angle Alpha changes.  You need to solve for Y and Z as a function of Alpha due to known tangential velocity V, plug those values into the equation for Theta, and take the derivative of Theta.  

This is a piecewise problem, since the solution is different for the straight section and the circular section.  For the straight section Y is a constant, and Theta is a function solely of distance along the x axis.  The angular velocity of the camera is a derivative of Theta, where theta is the inverse tangent of Y divided by X, and X is the integral of the speed V of the vehicle.

[zahc]

["With a constant tangential velocity V, the values of Y and Z are accelerating as the angle Alpha changes.  You need to solve for Y and Z as a function of Alpha due to known tangential velocity V, plug those values into the equation for Theta, and take the derivative of Theta. "]

I'm with you up to this point. However we know nothing about what you are calling Alpha. Unless you can related it back to Theta. I did, through the cosine and sine law, but it was nasty beyond being helpful.

What I did, since the problem specifies constant speed, is define Alpha through [VT/b], arc length divided by radius.

So my R vector was (a + Cos[VT/b])i + (Sin[VT/b])j.

Then I divided by |R|, rotated to form a tangential unit vector as a function of time, V,  and b. Then I found a unit vector tangent to the circle radius, and did V(that unit vector) dot (the other unit vector). Since I now had the component of V perpendicular to R then I divided by |R|.

Very nonelegant and probably wrong, but I'm done with Mechanics for this semester.