Chapter 7, Problem 94A

a.

To determine

Period of Moon in months for the given changes.

Answer to Problem 94A

New period of Moon is 0.707 months.

Explanation of Solution

Given:

Moon’s original period is one month.

Mass of Earth is doubled.

Formula used:

Time period is given as

  $T=2\pi \sqrt{\frac{r^3}{Gm}}$

Calculation:

It is given that mass is doubled but radius remains the same.

So, time period will be,

  $\begin{array}{c}T=2\pi \sqrt{\frac{r^3}{G\left(2m\right)}}\\ =\frac{1}{\sqrt{2}}\left(2\pi \sqrt{\frac{r^3}{Gm}}\right)\end{array}$

It is observed that time get reduced by a factor of $\frac{1}{\sqrt{2}}$ for the given changes.

Since period is 1 month so reduced period will be

  $\begin{array}{c}{T}_{new}=\frac{1}{\sqrt{2}}\times T\\ =\frac{1}{\sqrt{2}}\times 1\\ =0.707\text{months}\end{array}$

Conclusion:

Therefore, time period will be 0.707 months.

b.

To determine

Location of satellite.

Answer to Problem 94A

Location of satellite will be $r=1.26$ times the present orbital radius of Moon.

Explanation of Solution

Given:

Moon’s original period is one month.

Mass of Earth is doubled.

Formula used:

Time period is given as

  $T=2\pi \sqrt{\frac{r^3}{Gm}}$

Calculation:

Rearrange the time period formula,

  $\begin{array}{l}{\left(\frac{T}{2\pi }\right)}^2=\frac{r^3}{Gm}\\ r^3={\left(\frac{T}{2\pi }\right)}^{2}Gm\end{array}$

As, $m$ is doubled and other parameters are constant.

So, $r^3$ will be doubled.

Hence, $r=2^{\frac{1}{3}}=1.26$ times the present orbit radius of Moon.

Conclusion:

Therefore, the location of satellite will be $r=1.26$ times the present orbital radius of Moon

c.

To determine

Effect on length of year on Earth.

Answer to Problem 94A

The length of year on Earth will not change.

Explanation of Solution

Given:

Moon’s original period is one month.

Mass of Earth is doubled.

Formula used:

Orbital period of earth around the sun is given by relation

  $T=2\pi \sqrt{\frac{r^3}{Gm}}$

Here, $m$ is solar mass, $r$ is orbital radius of earth and $G$ is gravitational constant.

Calculation:

As, for the earth the revolution period around sun is

  $T=2\pi \sqrt{\frac{r^3}{Gm}}$

In this relation the period is independent of earth mass.

So on doubling the earth mass, there is no effect on the revolution period.

Hence length of year is unaffected.

Conclusion:

As revolution period of Earth is independent of Earth’s mass. So, the length of year on Earth will not change.