Chapter 1.5, Problem 58E

OTHER APPLICATIONS

Length of a Pendulum. A simple pendulum swings back and forth in regular time intervals. Grandfather clocks use pendulums to keep accurate time. The relationship between the length of a pendulum L and the period (time) T for one complete oscillation can be expressed by the function $L=k{T}^n$ where $k$ is a constant and $n$ is a positive integer to be determined. The data below were taken for different lengths of pendulums. Source: Gary Rockswold.

T(sec)	$L\left(ft\right)$
$1.11$	$1.0$
$1.36$	$1.5$
$1.57$	$2.0$
$1.76$	$2.5$
$1.92$	$3.0$
$2.08$	$3.5$
$2.22$	$4.0$

a. Find the value of $k$ for $n=1,2,\text{and}3,$ using the data for the $\text{4-ft}$ pendulum..

b. Use a graphing calculator to plot the data in the table and to graph the function $L=k{T}^n$ for the three values of $k$ (and corresponding values of $n$ ) found in part a. Which function best fits the data.

c. Use the best fitting function from part a to predict the period of a pendulum having a length of $\text{5}\text{ft}$.

d. If the length of the pendulum doubles, what happens to the period?

e. If you have a graphing calculator or computer program with a quadratic regression feature, use it to find a quadratic function that approximately fits the data. How does this answer compare with the answer to part b.