The displacement vector of a particle of mass m is given by$r(t)=\hat{i} A \operatorname{cos\omega t}+\hat{j} B \sin \omega t $a) show that the trajectory is an ellipse b) show that $F = -m\omega^2r$

Home » The displacement vector of a particle of mass m is given bya) show that the trajectory is an ellipse b) show that

The displacement vector of a particle of mass m is given by $r(t)=\hat{i} A \operatorname{cos\omega t}+\hat{j} B \sin \omega t$ a) show that the trajectory is an ellipse b) show that $F = -m\omega^2r$

CBSE | Class 11 | Laws of Motion | ncert exemplar | Physics

The displacement vector of a particle of mass m is given by $r(t)=\hat{i} A \operatorname{cos\omega t}+\hat{j} B \sin \omega t$ a) show that the trajectory is an ellipse b) show that $F = -m\omega^2r$

a)
$\begin{array}{l} r(t)=\hat{i} A \operatorname{coswt}+\hat{j} B \sin \omega t \\ x=A \cos \omega t \\ y=B \sin \omega t \\ x / A=\cos \omega t \\ y / B=\sin \omega t \end{array}$
Squaring both the equations and adding them,
$\begin{array}{l} x^{2} / A^{2}+y^{2} / B^{2}=\cos ^{2} \omega t+\sin ^{2} \omega t \\ x^{2} / A^{2}+y^{2} / B^{2}=1 \end{array}$
The above equation is of an ellipse and so the trajectory is an ellipse.
b) $x=A \cos \omega t$
$\begin{array}{l} v_{x}=d x / d t=-A \omega \sin \omega t \\ a_{x}=d v_{z} / d t=-A \omega^{2} \cos \omega t \\ y=B \sin \omega t \\ v_{y}=d y / d t=B \omega \cos \omega t \\ a_{y}=d v_{y} / d t=-B \omega^{2} \sin \omega t \\ a=a_{x} \hat{i}+a_{y} \hat{j} \end{array}$
Solving the above, we get
$a=-\omega^{2} r t$