Rod OA see the figure rotates about the fixed point O so tha

Rod OA (see the figure rotates about the fixed point O so that point A travels on a circle of radius r. Connected to point A is another rod AB of length L > r, and point B is connected to a piston. Show that the distance x between point O and point B is given by x = r cos theta + squareroot r^2 cos^2 theta + L^2 - r^2 where theta is the angle of rotation of rod OA.

Solution

using cosine law which is

a²=b²+c²-2bc cos theta  

we get

L²=r²+x²-2*r*x cos theta

L² - r²= x²-2*r*x cos theta + r²((cos²theta +sin²theta)-1)

L²-r²= x²-2*x*r cos theta +r²cos²theta +r²sin²theta^-r²

L²= x²-2*x*r cos theta +r²cos²theta + r²sin²theta

L²= (x- cos theta)² + r²(1-cos²theta)

L²-r²(1-cos²theta)=(x-r cos theta)²

Taking square root on both sides

x-r cos theta=sqrt(L²-r²(1-cos²theta))

x= r costheta + sqrt(L²-r²+r²cos²theta)

x= r cos theta + sqrt(r²cos²theta + L² - r²)