The displacement vector of a particle of mass m is given by r(t) = i A cos wt + j B sin wt. ​

Question

The displacement vector of a particle of mass m is given by r(t) = \(\hat{i}\) A cos wt + \(\hat{j}\) B sin wt.

(a) Show that the trajectory is an ellipse.

(b) Show that F = −mω²r.

Answer 1

 (a) x = A cos wt, y = B sin wt

\(\frac{x}{A}\) = cos wt           (1)

\(\frac{y}{B}\) = sin wt            (2)

Squaring & adding (1), (2)

\(\frac{x^2}{A^2}\) + \(\frac{y^2}{B^2}\) = 1

Which is the equation of ellipse

\(\frac{dr}{dt}\) = v = −\(\hat{i}\) w A sin wt + \(\hat{j}\) wB cos wt

\(\frac{dv}{dt}\) = a = −w²r; F = −mw²r