Chapter 5, Problem 6SP

The period of the moon’s orbit about the Earth is 27.3 days, but the average time between full moons is approximately 29.3 days. The difference is due to the motion of the Earth about the sun.

1. a. Through what fraction of its total orbital period does the Earth move in one period of the moon’s orbit?
2. b. Draw a sketch of the sun, the Earth, and the moon with the moon in the full moon condition. Then, sketch the position the moon would be in 27.3 days later, when the Earth is in its new position. If the moon is in the same position relative to Earth as it was 27.3 days earlier, is this a full moon?
3. c. How much farther would the moon have to go to reach the full moon condition? Show that this represents approximately an extra two days.

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 $0.075$.

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 $365.25\text{ days}$.

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

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

Conclusion:

Thus the fraction of total orbital period of earth through which it move in one period of the moon’s orbit is $0.075$.

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

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

The moon will not be a full moon yet if the moon is in the same position relative to Earth as it was $27.3\text{ days}$ earlier.

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.

$\text{Angle}=\frac{\text{total angle}}{\text{orbital period}}$ (1)

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

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

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

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

Find the net angle through the moon advances relative to the Earth.

$\begin{array}{c}\text{Net angle}=13.2°\text{/day}-1°\text{/day}\\ =12.2°\text{/day}\end{array}$

Find the total angle advanced by 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 yet if the moon is in the same position relative to Earth as it was $27.3\text{ days}$ earlier.

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 have 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 $333°$. Find the extra angle through which the moon have to go to reach the full moon condition.

$\begin{array}{c}\text{Extra angle}=360°-333°\\ =27°\end{array}$

Find the time taken to cover $27°$.

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

Conclusion:

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