Problem 808 - JEE Physics

Question

The radius of orbit of a planet is two times that of earth. The time period of planet is \(\ldots \ldots \ldots\) years. (A) \(4.2\) (B) \(2.8\) (C) \(5.6\) (D) \(8.4\)

Step-by-Step Solution

Answer

The time period of the planet is approximately \(2.8\) years.

Step 1: Recall Kepler's Third Law

We need to use Kepler's Third Law, which states that the square of the time period (T^2) of a planet is directly proportional to the cube of the semi-major axis (a^3) of its orbit; in a simplified form: \(T^2 \propto a^3\) We can also express this relationship using a constant of proportionality, denoted by 'k': \(T^2 = k * a^3\) Now, let's find the relationship between Earth's time period and the given planet's time period.

Step 2: Set up the proportionality equations for Earth and the given planet

Let's use subscript 1 for Earth and subscript 2 for the given planet. We can set up the following equations using Kepler's Third Law: \(T_1^2 = k * a_1^3\) \(T_2^2 = k * a_2^3\) Since the given planet's radius of orbit is twice that of Earth, we have: \(a_2 = 2 * a_1\) We also know that Earth has a time period of 1 year (T₁ = 1) because it takes 1 year for Earth to orbit around the sun.

Step 3: Find the ratio of time periods

Now we have all the information needed to find the time period of the given planet. We will find the ratio of \(T_2^2\) to \(T_1^2\): \(\frac{T_2^2}{T_1^2} = \frac{k*a_2^3}{k*a_1^3} \) Since the constants of proportionality (k) are equal, they cancel out: \(\frac{T_2^2}{T_1^2} = \frac{a_2^3}{a_1^3} \) Replace \(a_2\) with \(2*a_1\) and \(T_1\) with 1: \(\frac{T_2^2}{1^2} = \frac{(2*a_1)^3}{a_1^3} \)

Step 4: Solve for the time period of the given planet

Now, we just need to simplify and solve for the time period T₂: \(T_2^2 = \frac{(2^3 * a_1^3)}{a_1^3} \) \(T_2^2 = 2^3\) \(T_2^2 = 8\) Taking the square root of both sides: \(T_2 = \sqrt{8}\) \(T_2 \approx 2.83\) Based on the given options, the most suitable answer is (B) 2.8 years. So, the time period of the given planet is approximately 2.8 years.

Key Concepts

Planetary Orbits

In our solar system, planets move around the Sun in specific paths called orbits. These orbits are primarily elliptical but can be approximated as circles for simplicity when discussing general principles. An orbit is defined by its semi-major axis, which is the longest diameter of the ellipse. This axis plays a crucial role in determining the characteristics of a planet's orbit, such as its speed and the time it takes to complete one full orbit. Understanding planetary orbits helps us apply Kepler's Laws, which describe the motion of planets as they revolve around the Sun. These laws are vital for predicting how long a planet takes to orbit the Sun and comparing different planetary systems.

Time Period of Planets

The time period of a planet refers to how long it takes to complete one full orbit around the Sun. For Earth, this time period is one year. Using Kepler's Third Law, you can calculate the time period for any other planet if you know the length of its semi-major axis. According to the law, if you double the distance from the Sun, the time period doesn't just double. Instead, it increases in a more complex way, related to the cube of the distance. This allows scientists to predict the motion of planets with high accuracy. These calculations are crucial when comparing different planets within our solar system and beyond to gain a deeper understanding of their behavior.

Proportionality Equations

Proportionality equations are mathematical expressions that show a direct relationship between two or more variables. In the case of planetary motion, Kepler's Third Law offers a proportionality equation between the square of a planet's time period and the cube of its semi-major axis: \(T^2 \propto a^3\). This means if one variable changes, the other one changes in a predictable manner. The constant of proportionality in this case does not affect the relationship because it cancels out when comparing two planets. By using these equations, astronomers can determine unknown values, such as the time period of a planet, by understanding the relationship with its orbital radius.

Semi-Major Axis

The semi-major axis is the half-length of the longest diameter of an elliptical orbit. It is a fundamental parameter in describing an orbit's size. In the context of Kepler's Third Law, the semi-major axis serves as a critical factor in determining a planet's orbital period. The greater the semi-major axis, the longer it will take for a planet to complete one orbit around the Sun. Therefore, the semi-major axis not only helps in defining the shape and size of an orbit but also directly influences a planet's time period as per the law \(T^2 \propto a^3\). Knowing the semi-major axis enables scientists to calculate and predict the motion characteristics of planets, using mathematical models to simulate real cosmic movements.