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Thread: Finding the rise of an arc

  1. #1
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    Finding the rise of an arc

    Hi, I need help with a problem the best way of describing it is as follows

    I have a pendulum (100cm long) attached to it is a piece of string with a small weight at the end.
    The pendulum is pushed back moving the weight 10cm.
    How do I calculate the amount the pendulum has risen on its arc?

    Does anyone have a formula to work this out.

    Any help very much appreciated
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  2. #2
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    Re: Finding the rise of an arc

    Hello, soapes!

    I have a pendulum 100cm long with a small weight at the end.
    The pendulum is pushed back moving the weight 10cm.
    How do I calculate the amount the pendulum has risen on its arc?

    Code:
                    O
                    *
                   /|\
                  /@| \
             100 /  |  \
                /   |   \
               /    |C   \
            B * - - + - - *
                *   |h  *
               10   *
                    A
    The pendulum is pivoted at $\displaystyle O$.
    The weight is at rest at $\displaystyle A\!:\:OA = 100$ cm.

    It is moved 10 cm to $\displaystyle B.$
    Arc $\displaystyle \overline{AB} = 10$ cm..$\displaystyle OB = 100$ cm.
    Let $\displaystyle \theta = \angle AOB.$
    We want: $\displaystyle h = AC.$

    We have:.$\displaystyle s = r\:\!\theta \quad\Rightarrow\quad 10 = 100\:\!\theta \quad\Rightarrow\quad \theta = \tfrac{1}{10}$

    Then:.$\displaystyle \cos\theta = \frac{OC}{OB} \quad\Rightarrow\quad OC = OB\cos\theta = 100\cos\left(\tfrac{1}{10}\right)$

    Therefore:.$\displaystyle h \:=\:OA - OC \:=\:100 - 100\cos(\tfrac{1}{10}) $

    . . . . . . . . $\displaystyle h \:=\:0.499583472 \:\approx\:0.5\text{ cm}$

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  3. #3
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    Re: Finding the rise of an arc

    Thanks for your help, I'm a bit rusty at maths so still trying to find my feet.

    Another quick question, would the pendulums rise be constant, for example we know that 10cm movement=0.5cm rise, so would a 5cm movement= 0.25cm?

    Thanks again.
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  4. #4
    MHF Contributor

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    Re: Finding the rise of an arc

    No, as Soroban's response shows, the relation is not linear.
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