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Math Help - Pendulum Problem using derivative of sine/cosine

  1. #1
    Newbie mockingjay95's Avatar
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    Pendulum Problem using derivative of sine/cosine

    Pendulum completes swing (back and forth) each 6s. As it swings, the distance, d, in cm, depends on time t, in seconds. At t=1.3 d is at its max of 110cm. Its min is 50cm. D is sinusoidal function of t.
    I got the equation y=80+60cospi/3(t-1.3) using the methods from the book (find the sine axis, amplitude, high point and B), is there a better way/ faster way to do it?
    The second question asks for the fastest pendulum swing, and where it is swinging its fastest. I don't understand how to solve for this question.
    For the third question what is the first positive value of t at which the pendulum is swinging 0cm/s and where is the pendulum at this time- do you find the third derivative y'' and then solve for y''=0? then put the value t into the main equation y?
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  2. #2
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    Re: Pendulum Problem using derivative of sine/cosine

    You will need to find the formula for speed. Then maximize that function.

    For the third question, set the speed function to 0 and solve for the minimum positive value of t.

    For reference, the speed function is the magnitude of the velocity function.
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  3. #3
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Pendulum Problem using derivative of sine/cosine

    T=2\pi \sqrt{l/g}=6 -> L = 894.26 cm (length of pendulum), g = 980.665 cm/s/s

    \theta (\text{t})\text{=}\text{c1}* \text{Cos}\left[\frac{\sqrt{g} t}{\sqrt{l}}\right]+\text{c2}* \text{Sin}\left[\frac{\sqrt{g} t}{\sqrt{l}}\right]

    c1=0.0069762
    c2=0.03282

    theta = angle of pendulum with vertical



    max velocity = 31.421 cm/s at theta = 0 (vertical position)

    1st velocity =0 at t=1.3

    You can easily transfer theta(t) to horizontal d(t).
    Attached Thumbnails Attached Thumbnails Pendulum Problem using derivative of sine/cosine-pendulum.png  
    Last edited by MaxJasper; October 8th 2012 at 10:23 PM.
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    Newbie mockingjay95's Avatar
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    Re: Pendulum Problem using derivative of sine/cosine

    Can you clarify the formula with c1 and c2?
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Pendulum Problem using derivative of sine/cosine

    Attached Thumbnails Attached Thumbnails Pendulum Problem using derivative of sine/cosine-pendulum2.png  
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