Question

The profit of a cam in a particular zone is given by x = $\sqrt{3}cos\theta$ and y = sin$\theta$. The normal to the cam profile at $\theta =\frac{\pi }{4}$ is at an angle (with respect to x axis):

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. 0

Answer 1

The correct option is C $\frac{\pi }{3}$

x = $\sqrt{3}cos\theta$

y = sin $\theta$

Slope of cam profile at an angle $\theta$ is

$\frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{1}{\left(\frac{dx}{d\theta }\right)}$

$=\frac{cos\theta }{-\sqrt{3}sin\theta }=\frac{-cot\theta }{\sqrt{3}}$

Solpe of normal to cam profile at angle $\theta$ is

m = $\frac{-1}{\left(\frac{dy}{dx}\right)}=\sqrt{3}tan\theta$

$m=\sqrt{3}tan\left(\frac{\pi }{4}\right)$

$m=\sqrt{3}$

Angle of Normal with x-axis is

$\alpha =ta{n}^{-1}\left(m\right)=ta{n}^{-1}\left(\sqrt{3}\right)$

$\alpha ={60}^{\circ }=\frac{\pi }{3}$