
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, $\displaystyle y = me^{ft} $ and differentiate with respect to t twice, yielding $\displaystyle y'' $. Plug these into your differential equation and solve for f (you should get two values). With two possibilities for f, the general solution is
$\displaystyle y(t) = m_{1}e^{f_{1}t} + m_{2}e^{f_{2}t} $
To write in the sin + cos form, use the Euler identity:
$\displaystyle e^{\pm i \phi} = \cos{\phi} \pm \sin{\phi} $
For the last one use the trig identity:
$\displaystyle \sin{(a \pm b)} = \sin{a} \cos{b} \pm \cos{a} \sin{b} $

Re: Harmonic Motion
Part 2
Use the formula $\displaystyle \vec{F} = m \frac{d^{2} \vec{r}}{dt^{2}} $, where $\displaystyle \vec{r} = r \hat{r} $. But be careful! You have differentiate $\displaystyle \hat{r} $ as well as $\displaystyle r$, since in polar coordinates the directional vectors change depending on position, unlike in Cartesian coordinates.
So write out the directional vectors in Cartesian coordinates.
$\displaystyle \hat{r} = \left( \cos{\theta}, \sin{\theta} \right) , \hat{\theta} = \left( \sin{\theta}, \cos{\theta} \right) $
Then the first time derivative (denoted by one dot) is
$\displaystyle \frac{d \vec{r}}{dt} = \dot{\vec{r}} = \dot{r} \hat{r} + r \dot{\hat{r}} $
$\displaystyle = \dot{r} \hat{r} + r \left( \dot{\theta} \sin{\theta}, \dot{\theta} \cos{\theta} \right) $
$\displaystyle = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta} $
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