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  • The profile of a cam in a particular zone is given by <img alt="\mathrm{ x=\sqrt{3} \cos \theta}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20x%3D%5Csqrt%7B3%7D%20%5Ccos%20%

The profile of a cam in a particular zone is given by \mathrm{ x=\sqrt{3} \cos \theta} and \mathrm{ y=\sin \theta}. The normal to the cam profile at

\mathrm{\theta=\frac{\pi}{4}} is at angle (with respect to x axis):

Option: 1

\frac{\pi}{4}


Option: 2

\frac{\pi}{2}


Option: 3

\frac{\pi}{3}


Option: 4

0


Answers (1)

best_answer

              \mathrm{ \begin{aligned} & x=\sqrt{3} \cos \theta \\ & y=\sin \theta \end{aligned} }
Slope of cam profile at an angle \mathrm{\theta} is

\mathrm{ \begin{aligned} \frac{d y}{d x} & =\frac{d y}{d \theta} \times \frac{1}{\left(\frac{d x}{d \theta}\right)} \\ & =\frac{\cos \theta}{-\sqrt{3} \sin \theta}=\frac{-\cot \theta}{\sqrt{3}} \end{aligned} }

Slope of Normal to cam profile at angle \mathrm{\theta} is

\mathrm{ m=\frac{-1}{\left(\frac{d y}{d x}\right)}=\sqrt{3} \tan \theta }

\mathrm{ \begin{aligned} & m=\sqrt{3} \tan \left(\frac{\pi}{4}\right) \\ & m=\sqrt{3} \end{aligned} }

Angle of Normal with x-axis is

\mathrm{ \begin{aligned} & \alpha=\tan ^{-1}(\mathrm{~m})=\tan ^{-1}(\sqrt{3}) \\ & \alpha=60^{\circ}=\frac{\pi}{3} \end{aligned} }

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