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.

localid="1646318957850" $T=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$

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 localid="1646318988440" $l$changes the period localid="1646318997671" $T$.

(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 $l$localid="1646319009284" $l$by factors of $2,3$and $4$changes the period $T$.

Short 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.

Step-by-step solution

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 Step1 The graph of the function T=Tl+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=T2l 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.