Harmonic Motion

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August 6th 2012, 09:44 AM
Mathsnewbie

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= α-βcos(2πt)
Theta=μ-vsin(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
August 6th 2012, 10:49 AM
JohnDMalcolm

Re: Harmonic Motion

Part 1
Take your trial equation, $y = me^{ft}$ and differentiate with respect to t twice, yielding $y''$ . Plug these into your differential equation and solve for f (you should get two values). With two possibilities for f, the general solution is

$y(t) = m_{1}e^{f_{1}t} + m_{2}e^{f_{2}t}$

To write in the sin + cos form, use the Euler identity:

$e^{\pm i \phi} = \cos{\phi} \pm \sin{\phi}$

For the last one use the trig identity:

$\sin{(a \pm b)} = \sin{a} \cos{b} \pm \cos{a} \sin{b}$
August 6th 2012, 11:06 AM
JohnDMalcolm

Re: Harmonic Motion

Part 2
Use the formula $\vec{F} = m \frac{d^{2} \vec{r}}{dt^{2}}$ , where $\vec{r} = r \hat{r}$ . But be careful! You have differentiate $\hat{r}$ as well as $r$ , since in polar coordinates the directional vectors change depending on position, unlike in Cartesian coordinates.

So write out the directional vectors in Cartesian coordinates.

$\hat{r} = \left( \cos{\theta}, \sin{\theta} \right) , \hat{\theta} = \left( -\sin{\theta}, \cos{\theta} \right)$

Then the first time derivative (denoted by one dot) is

$\frac{d \vec{r}}{dt} = \dot{\vec{r}} = \dot{r} \hat{r} + r \dot{\hat{r}}$
$= \dot{r} \hat{r} + r \left( -\dot{\theta} \sin{\theta}, \dot{\theta} \cos{\theta} \right)$
$= \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$

See if you can do the second time derivative, which will give you an expression to compute the angular component of the force.
August 6th 2012, 02:55 PM
Mathsnewbie

Re: Harmonic Motion

Thank you!!