
Harmonic Motion
Hey guys completely lost how to do this. I'd appreciate any help or any website that may help with it.
1. Solve the equation y'' + w^2.y = 0 by making the Ansatz y=m.e^f.t where m and f are some constants. Verify the solution to the harmonic motion can be expressed in any of the following three forms,
y(t) =B1e^iwt+B2e^iwt
=C1Sinwt+C2Coswt
=ASin(wt+theta)
2. A Robot is programmed so that the particle P of Mass m, describes the path
r= αβcos(2πt)
Theta=μvsin(2πt)
where alpha, beta, mu and v are positive constants. Determine the polar components of force exerted on p by the robots Jaws at time t=2s

Re: Harmonic Motion
Part 1
Take your trial equation, and differentiate with respect to t twice, yielding . Plug these into your differential equation and solve for f (you should get two values). With two possibilities for f, the general solution is
To write in the sin + cos form, use the Euler identity:
For the last one use the trig identity:

Re: Harmonic Motion
Part 2
Use the formula , where . But be careful! You have differentiate as well as , since in polar coordinates the directional vectors change depending on position, unlike in Cartesian coordinates.
So write out the directional vectors in Cartesian coordinates.
Then the first time derivative (denoted by one dot) is
See if you can do the second time derivative, which will give you an expression to compute the angular component of the force.

Re: Harmonic Motion