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

The displacement vector of a mass m is given by r (t) = hati A cos ome

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