The displacement vector of a particle of mass m is given by  r(t)=ȋ A cos $\mathit{w}\mathit{t}+$ ĵ B sinwt.

(a) Show that the trajectory is an ellipse.

(b) Show that F= -mw²/r

<p><span lang="EN-IN">This is a Long Answer type Questions as classified in NCERT Exemplar</span></p><p><strong><span lang="EN-IN">Explanation-&nbsp;</span></strong><span lang="EN-IN">displacement vector of particle is r (t)=? Acos<span contenteditable="false"> <math> <mi>w</mi> <mi>t</mi> <mo>+</mo> </math> </span></span><span lang="EN-IN">? Bsinwt</span></p><p><span lang="EN-IN">X=Acoswt</span></p><p><span lang="EN-IN">x/A= coswt………1</span></p><p><span lang="EN-IN">displacement along y axis is </span></p><p><span lang="EN-IN">y=Bsinwt</span></p><p><span lang="EN-IN">y/B= sinwt……….2</span></p><p><span lang="EN-IN">squaring and then adding eqn1 and 2 we get </span></p><p><span lang="EN-IN">x²/A²+y²/B²=cos²wt+sin²wt =1</span></p><p><span lang="EN-IN">this is an equation of ellipse. Therefore trajectory of particle is an ellipse.</span></p><p><span lang="EN-IN">b)v= dr/dt= id/dt (Acoswt)+jd/dt (Bsinwt)</span></p><p><span lang="EN-IN">&nbsp;? [A (-sinwt)w]+? [B (coswt).w]</span></p><p><span lang="EN-IN">= -? Awsinwt+? Bwcoswt</span></p><p><span lang="EN-IN">Acceleration a= dv/dt</span></p><p><span lang="EN-IN">So a= -? Awd/dt (sinwt)+? Bw [-sinwt]w</span></p><p><span lang="EN-IN">=? Aw²coswt-? Bsinwt</span></p><p><span lang="EN-IN">= -w²r</span></p><p><span lang="EN-IN">So force acting on the particle f=ma=-mrw²</span></p>

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