# Math Help - Finding the rise of an arc

1. ## 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

2. ## 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 $O$.
The weight is at rest at $A\!:\:OA = 100$ cm.

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

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

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

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

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

3. ## 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.

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

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