An object with mass m undergoes simple harmonic motion, following 2 perpendicular directions, described by the equations:
x=a cos (wt), a>0,
y=b cos (2wt), b>0
a) find the equation of the trajectory
b) find the speed at any given time (so having t as a variable)
c) the maximum force F which acts on the object at any given time (again, having t as a variable).
The Attempt at a Solution
So far: a) from x=a cos (wt) we get cos (wt)=x/a; in the y equation, we can expand as follows:
cos (2wt)=cos^2 (wt) - sin^2 (wt). We also know that for any real x we have cos^2 (x) +sin^2(x)=1, therefore cos (2wt)=cos^2 (wt) - sin^2 (wt)=cos^2 (wt) + cos^2(wt)-1; therefore y=b (2cos^2 (wt) -1 )=b( x^2/a^2 -1 ), which is the equation of the trajectory.
Now for b) and c), I'm not quite sure how to use what I have. I differentiated the x and y equation from the beginning, differentiated the trajectory and somehow I need to combine them. I suppose the idea from b) applies to c).
I would be grateful if you could give me some hints :)
Have a great day,
PS: If it is not clear, I could rewrite using latex.