# Need help Mathematical Models!

• Nov 27th 2006, 03:21 AM
Need help Mathematical Models!
The period (the time for one complete oscillation) of a pendulum is directly proportional to the square root of the length of the pendulum, and a pendulum of length 8 ft has a period of 2 seconds. Find a mathematical model expressing the period of a pendulum as a function of its length:

Why is the answer equal to $\displaystyle P(x) = \sqrt {\frac{x}{2}}$?
and thanks
• Nov 27th 2006, 03:50 AM
topsquark
Quote:

The period (the time for one complete oscillation) of a pendulum is directly proportional to the square root of the length of the pendulum, and a pendulum of length 8 ft has a period of 2 seconds. Find a mathematical model expressing the period of a pendulum as a function of its length:

Why is the answer equal to $\displaystyle P(x) = \sqrt {\frac{x}{2}}$?
and thanks

I presume x is the length of the pendulum.

We know that P(x) is proportional to $\displaystyle \sqrt{x}$, so let
$\displaystyle P(x) = k \sqrt{x}$ where k is a constant.

Since a pendulum of 8 ft has a period of 2 seconds:
$\displaystyle 2 = k \sqrt{8}$

$\displaystyle k = \frac{2}{\sqrt{8}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}}$

Thus
$\displaystyle P(x) = \frac{1}{\sqrt{2}} \sqrt{x} = \sqrt{\frac{x}{2}}$

-Dan