The displacement vector of a particle of mass m is given by r (t) = `hati A cos omega t + hatj B sin omega t`.
(a) Show that the trajectory is an ellipse.
(b) Show that F = `-m omega^(2)r`.

Explanation

Here (a) ` vec(r) (t) = hati A cos omegat + hatj B sin omegat, :. x = A cos omegat, y = B sin omegat `
` x^(2)/(A^(2)) + y^(2)/(B^(2)) cos^(2) omegat + sin^(2) omegat = 1` which is the equation of an ellipse
`:.` The trajectory of the particle is elliptical
(b) Now ` vec(upsilon) =vec (dr)/(dt) = -hatiomega A sin omegat + hatjomega Bcos omegat `
` vec(a) = vec (d upsilon)/(dt) = hati omega^(2) A cos omegat - hatj omega^(2) B sin omegat = - omega^(2) [hati A cos omegat + hatjBsin omegat] = - omega^(2) vec(r) `
` vec(F) = m vec (a) = - m omega^(2) vec(r) ` .