Q. 88 - Precalculus Enhanced with Graphing Utilities

Question

Period of a Pendulum The period $T$(in seconds) of a simple pendulum is a function of its length $l$ (in feet) defined by the equation.

            $T=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$

where $g\approx 32.2$feet per second per second is the acceleration of gravity.

(a) Use a graphing utility to graph the function $T=T\left(l\right)$.

(b) Now graph the functions $T=T\left(l+1\right),T=T\left(l+2\right)$ and $T=T\left(l+3\right)$

(c) Discuss how adding to the length $l$ changes the period $T$.

(d) Now graph the functions $T=T\left(2l\right),T=T\left(3l\right)$ and $T=T\left(4l\right)$.

(e) Discuss how multiplying the length $l$$l$ by factors of $2,3$and $4$changes the period $T$.

Step-by-Step Solution

Answer

$\left(a\right)$ The graph for the function $T=T\left(l\right)$ is,

$\left(b\right)$The graph of the function $T=T\left(l+1\right)$is,

The graph of the function $T=T\left(l+2\right)$ is,

The graph of the function $T=T\left(l+3\right)$ is,

$\left(c\right)$ Adding to the length increases the time period as time period is directly proportional to the length.

$\left(d\right)$ The graph of the function $T=T\left(2l\right)$ is,

 

The graph of the function $T=T\left(3l\right)$ is,

The graph of the function $T=T\left(4l\right)$ is,

$\left(e\right)$ The time period increases as it is directly proportional to the length. 

Part a Step 1 Let us consider the time period of a simple pendulum in terms of its length.

The graph of the function$T=T\left(l\right)$ is,

Part b Step 1 The graph of the function T = T l + 1 is,

The graph of the function $T=T\left(l+2\right)$ is,

The graph of the function $T=T\left(l+3\right)$ is,

Part c Step 1 Explanation for adding the length l changes the period T .

Adding to the length increases the time period as time period is directly proportional to the length.

Part d Step 1 The graph of the function T = T 2 l is,

The graph of the function $T=T\left(3l\right)$ is,

The graph of the function $T=T\left(4l\right)$ is,

Part e Explanation for the changes in period T by multiplying the length with different length :

The time period increases as it is directly proportional to the length.