A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by
a. Eliminate t from these equations to show that the path of the particle in the x-y plane is an ellipse.
b. Calculate the kinetic and potential energy at a point on the particle’s trajectory. Show that the ellipse is a path of constant total energy, and show that the total energy is given by the sum of the separate energies of the x and y oscillations.
Equation of an ellipse: (x/a)^2+(y/b)^2=1
KE = 1/2=mv^2
PE = ??
The Attempt at a Solution
a) I rearranged the parametric eqns. like so: x/a=sinwt and y/b=coswt, squared both sides and using trig identities eliminated the term (sin^2wt+cos^2wt) leaving 1 on the right side and therefore getting the equation of an ellipse: (x/a)^2+(y/b)^2=1
b)This is where I'm having some troubles... I am pretty sure that I can calculate the KE of the particle by looking at:
but I don't know how to come up with the equations for the potential energy.
Regrading the fact that the ellipse is a path of constant total energy, I am thinking of showing that when the particle is on the x axis it means that y=0 thus eliminating the y component of the energy, same goes for when its on the y axis. which essentially can be modelled with a cosine function where energy is always constant but changes from component to component.
I'd appreciate any help or some input on my attempt at the solution.