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The following table shows the duration of a year on some unknown planets of equal mass.

\begin{tabular}{|c|c|}
\hline Planet & Duration of Year \\
\hline[tex]$Y$[/tex] & 224 days \\
\hline[tex]$Z$[/tex] & 320 days \\
\hline
\end{tabular}

Which of the following best explains which planet is closer to the sun?

A. Planet [tex]$Y$[/tex], because gravity is directly proportional to the square of distance.

B. Planet [tex]$Z$[/tex], because gravity is directly proportional to the square of distance.

C. Planet [tex]$Z$[/tex], because the sun exerts a greater gravitational force on it than on planet [tex]$Y$[/tex].

D. Planet [tex]$Y$[/tex], because the sun exerts a greater gravitational force on it than on planet [tex]$Z$[/tex].



Answer :

GinnyAnswer
To determine which planet is closer to the sun, we can use Kepler's third law of planetary motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (R), which represents the distance from the sun. Mathematically, this is expressed as:

[tex]\[ T^2 \propto R^3 \][/tex]

Given the durations (orbital periods) of the two planets:
- Planet Y has an orbital period of 224 days.
- Planet Z has an orbital period of 320 days.

Let's analyze these durations step-by-step:

1. Understanding the Proportionality: According to Kepler’s third law:
- A shorter orbital period (T) means that the planet has a smaller semi-major axis (R), implying it is closer to the sun.
- Conversely, a longer orbital period (T) means a larger semi-major axis (R), implying the planet is farther from the sun.

2. Comparison of Year Durations:
- Planet Y: 224 days
- Planet Z: 320 days

Planet Y has a shorter period compared to Planet Z.

3. Conclusion from the Orbital Periods:
- Since Planet Y has a shorter year duration (224 days) compared to Planet Z (320 days), Planet Y must be closer to the sun.

4. Gravitational Force Consideration:
- The sun exerts a greater gravitational force on planets that are closer to it.
- Therefore, the planet closer to the sun (Planet Y) will experience a stronger gravitational pull from the sun.

Considering all these points, the correct explanation is:

Planet Y, because the sun exerts a greater gravitational force on it than on planet Z.

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