This is a problem where we are supposed to design a cam..

GIVEN $\displaystyle s = F(theta) = L/2*(1-cos(2*pi*theta/beta))$

For the same harmonic motion describe above, show that for a fiven value of C and L, the cam angle must be limited by the relationship..

$\displaystyle beta^2 >=2*pi^2*(L/(C+L))$

in order to avoid pointing or cusp of the cam surface.

I know a cusp or point occurs on the cam surface when both $\displaystyle dx/dtheta=0$ and $\displaystyle dy/dtheta=0$

I also know that

$\displaystyle x = [ C + f(theta)]*cos(theta) - f`(theta)*sin(theta)$

so...

$\displaystyle dx/dtheta= - [C + f(theta) +f``(theta)]*sin(theta)$

So plugging in

$\displaystyle -sin(theta)*[C + L/2 -(L/2)*cos(2*pi*theta/beta) + ((4*L*pi^2)/(2*beta^2))*cos(2*pi*theta/beta)=0$


I just need help simplifying so i can get the relation above