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Math Help - Harmonic Motion

  1. #1
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    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
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  2. #2
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    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}
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  3. #3
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    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.
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  4. #4
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    Re: Harmonic Motion

    Thank you!!
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