Question

A particle travels clockwise on a circular path of diameter​ R, monitored by a sensor on the circle at point​ P; the other endpoint of the diameter on which the sensor lies is Q. Let theta be the angle between the diameter PQ and the line from the sensor to the particle. Let c be the length of the chord from the​ particle's position to Q. Complete parts a and b below.a. Calculate d theta/dc b. Evaluate d theta/dc with c=0

Answer 1

We make a graphic of this problem to define the angle.

The angle we can calculate through triangle relation, that is,

$sin\theta = \frac{c}{QP}\\sin\theta = \frac{c}{R}\\\theta=sin^{-1}\frac{c}{R}$

With this function we should only calculate the derivate in function of c

$\frac{d\theta}{dc} = \frac{1}{\sqrt{1-\frac{c^2}{R^2}}}(\frac{c}{R})'\\\frac{d\theta}{dc} = \frac{1}{\sqrt{R^2-c^2}}$

That is the rate of change of $\theta$.

b) At this point we need only make a substitution of 0 for c in the equation previously found.

$\frac{d\theta}{dc}\big|_{c=0} = \frac{1}{\sqrt{R^2-0}}\\\frac{d\theta}{dc}\big|_{c=0} = \frac{1}{R}$

Hence we have finally the rate of change when c=0.