Chapter 5, Problem 6SP

(a)

To determine

The fraction of total orbital period of earth through which it move in one period of the moon’s orbit.

Answer to Problem 6SP

The fraction of total orbital period of earth through which it move in one period of the moon’s orbit is $7.5\%$.

Explanation of Solution

Given Info: The period of moon’s orbit about the Earth is $27.3\text{ days}$.

The period of orbital motion of Earth is $365.25\text{ days}$. Therefore fraction of total orbital period of earth through which it move in one period of moon’s orbit is obtained by taking the ratio of period of moon’s orbit about earth to total orbital period of earth.

Write the expression for the fraction of total orbital period of Earth through which it move in one period of the moon’s orbit.

$\begin{array}{c}\frac{\text{Period of moon's orbit about Earth}}{\text{Period of orbital motion of Earth}}=\frac{27.3\text{ days}}{365.25\text{ days}}\\ =0.075\end{array}$

Write the fraction $0.075$ to percentage.

$\begin{array}{c}0.075=0.075\times 100\\ =7.5\%\end{array}$

Conclusion:

Thus, the fraction of total orbital period of earth through which it move in one period of the moon’s orbit is $7.5\%$.

(b)

To determine

The sketch of the sun, the Earth and the moon with the moon in the Full moon condition and also the sketch of the position where the moon would be in $27.3\text{ days}$ later when the Earth is in its new position and to determine whether it will be a full moon if the moon is in the same position relative to Earth as it was $27.3\text{ days}$ earlier.

Answer to Problem 6SP

The sketch of the sun, the Earth and the moon with the moon in the Full moon condition is  given in figure 1.and the sketch of the position where the moon would be in $27.3\text{ days}$ later when the Earth is in its new position is given in figure 2.

The moon will not be a full moon since the apparent advance of the moon in $27.3\text{days}$ is only $333°$.

Explanation of Solution

When the moon is full, it is on the opposite side of the Earth from the sun. The sketch of the sun, the Earth and the moon with the moon in the Full moon condition is plotted in figure 1.

Figure 1

Write the expression for the angle through which the moon or the Earth advances in one day.

$\theta =\frac{{\theta }_T}{T}$ (1)

Here,

$\theta$ is the angle through which moon or earth advances in one day

${\theta }_T$ is the total angle advanced by earth or moon in one day

$T$ is the time period

Substitute $360°$ for $Q_T$ and $27.3\text{ days}$ for $T$ in equation (1) to find the angle through which the moon advances in one day.

$\begin{array}{c}\theta =\frac{360°}{27.3\text{ days}}\\ =13.2°\text{/day}\end{array}$

Substitute $360°$ for $Q_T$ and $\text{365}\text{.25 days}$ for $T$ in equation (1) to find the angle through which the earth advances in one day.

$\begin{array}{c}\theta =\frac{360°}{365.25\text{ days}}\\ \approx 1°\text{/day}\end{array}$

Find the apparent advance of the moon relative to sun earth system in one day.

$\begin{array}{c}{\theta }_{\text{net}}=13.2°\text{/day}-1°\text{/day}\\ =12.2°\text{/day}\end{array}$

Here,

${\theta }_{\text{net}}$ is the apparent advance of the moon relative to sun earth system in one day

Find the apparent advance of the moon in $27.3\text{ days}$.

$\begin{array}{c}\left(12.2°\text{/day}\right)\times \left(27.3\text{ days}\right)=333°\\ \ne \text{360 }°\end{array}$

The moon is not full moon yet.

The sketch of the position where the moon would be in $27.3\text{ days}$ later when the Earth is in its new position is drawn in figure (2).

Figure 2

Conclusion:

Thus, the sketch of the sun, the Earth and the moon with the moon in the Full moon condition is plotted in figure 1 and the sketch of the position where the moon would be in $27.3\text{ days}$ later when the Earth is in its new position is plotted in figure 2. The moon will not be a full moon since the apparent advance of the moon in $27.3\text{days}$ is only $333°$.

(c)

To determine

The extra angle through which the moon have to go to reach the full moon condition and to show it represents approximately an extra two days.

Answer to Problem 6SP

The extra angle through which the moon has to go to reach the full moon condition is $27°$ and it takes $2.2\text{ days}$ more which is approximately extra two days.

Explanation of Solution

From part (b), the angle advanced by moon in $27.3\text{ days}$ is $333°$.

Therefore the extra angle that it can have to go to reach the full moon condition is obtained by subtracting apparent angle advanced by moon to total angle.

Find the extra angle for the moon to reach full moon condition.

$\begin{array}{c}\Delta \theta =360°-333°\\ =27°\end{array}$

Here,

$\Delta \theta$ is the extra angle for the moon to reach full moon condition.

Write the expression for the time taken to reach full moon condition.

$t=\frac{\Delta \theta }{{\theta }_{\text{net}}}$

Here,

$t$ is the time taken to reach full moon condition

Substitute $27°$ for $\Delta \theta$ and $12.2°\text{/day}$ for ${\theta }_{\text{net}}$ in the above equation to get $t$.

$\begin{array}{c}t=\frac{27°}{12.2°\text{/day}}\\ =2.2\text{ days}\\ \approx 2\text{ days}\end{array}$

Conclusion:

Thus, the extra angle through which the moon has to go to reach the full moon is condition $27°$ and it takes approximately an extra two days.