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3-2.Motion in Plane
medium
A particle moves in $x-y$ plane according to rule $x = {\rm{asin}}\omega t$ and $\;y = {\rm{acos}}\omega t$. The particle follows
Aan elliptical path
Ba circular path
Ca parabolic path
Da straight line path inclined equally to $x$ and $y$ axis.
(AIPMT-2010)
Solution
$\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,x = a\,\sin \,\omega t\,\,\,or\,\frac{x}{a} = \sin \,\omega t\,\,\,\,\,…\left( i \right)\\
\,\,\,\,\,\,\,\,\,y = \,a\,\cos \,\omega t\,\,or\,\frac{y}{a} = \,\cos \,\omega t\,\,\,\,…\left( {ii} \right)\\
Squaring\,and\,adding,\,we\,get\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\,\,\,\left( {\,{{\cos }^2}\,\omega t + \sin \omega t = 1} \right)\\
or\,\,\,{x^2} + {y^2} = {a^2}\\
This\,is\,the\,equation\,of\,a\,circle.\,Hence\\
particle\,follows\,a\,circular\,path.
\end{array}$
\,\,\,\,\,\,\,\,\,\,x = a\,\sin \,\omega t\,\,\,or\,\frac{x}{a} = \sin \,\omega t\,\,\,\,\,…\left( i \right)\\
\,\,\,\,\,\,\,\,\,y = \,a\,\cos \,\omega t\,\,or\,\frac{y}{a} = \,\cos \,\omega t\,\,\,\,…\left( {ii} \right)\\
Squaring\,and\,adding,\,we\,get\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\,\,\,\left( {\,{{\cos }^2}\,\omega t + \sin \omega t = 1} \right)\\
or\,\,\,{x^2} + {y^2} = {a^2}\\
This\,is\,the\,equation\,of\,a\,circle.\,Hence\\
particle\,follows\,a\,circular\,path.
\end{array}$
Standard 11
Physics
