Chapter 1.11, Problem 30E

(A)

To determine

To find:

The equation given that the statement

Answer to Problem 30E

The equation is $T=kx$

Explanation of Solution

Given:

  $T$ varies directly as $x$

Concept used:

If the quantities $x$ and $y$ are related by an equation

  $y=kx$

For some constant $k\ne 0$

That is y varies directly as $x$ or y is directly proportional to $x$

The constant $k$ is called constant proportionality

Calculation:

If the quantities $P$ and $l$ are related by an equation

  $P\alpha \sqrt{l}$

  $P=k\sqrt{l}\mathrm{................}\left(1\right)$

For some constant $k\ne 0$

That is $P$ varies directly as $\sqrt{l}$ or $P$ is directly proportional to $\sqrt{l}$

The constant $k$ is called constant proportionality

Where P is the period of a pendulum

  $l$ is the length of the pendulum

(B)

To determine

To find:

The equation of double the period that the statement

Answer to Problem 30E

The equation is $l_2=4l_1$

Explanation of Solution

Given:

  $P$ is directly proportional to $\sqrt{l}$

Concept used:

If the quantities $x$ and $y$ are related by an equation

  $y=kx$

For some constant $k\ne 0$

That is y varies directly as $x$ or y is directly proportional to $x$

The constant $k$ is called constant proportionality

Calculation:

If the quantities $P$ and $l$ are related by an equation

  $P\alpha \sqrt{l}$

  $P=k\sqrt{l}\mathrm{................}\left(1\right)$

For some constant $k\ne 0$

That is $P$ varies directly as $\sqrt{l}$ or $P$ is directly proportional to $\sqrt{l}$

The constant $k$ is called constant proportionality

Where P is the period of a pendulum

  $l$ is the length of the pendulum

That is

  $P_2=2P_1\mathrm{...........................}\left(2\right)$

  $\begin{array}{l}k\sqrt{l_2}=2k\sqrt{l_1}\\ \sqrt{l_2}=2\sqrt{l_1}\\ l_2=4l_1\end{array}$

The new length has to quadruple the first length