Problem 87 - Master Resource Book in JEE Main Physics

Question

The period of revolution of planet \(A\) around the sun is 8 times that \(B\). The distance of a from the sun is how many times greater than that of \(B\) from the sun? (a) 2 (b) 3 (c) 4 (d) 5

Step-by-Step Solution

Answer

The distance of planet A from the sun is 4 times greater than that of planet B.

Step 1: Understanding Kepler's Third Law

Kepler's Third Law states that the square of the period of revolution of a planet (T) is proportional to the cube of the semi-major axis of its orbit (R). This can be written as \( T^2 \propto R^3 \). This implies \( \frac{T_A^2}{T_B^2} = \frac{R_A^3}{R_B^3} \) for two planets A and B.

Step 2: Setting Up the Proportion

Given that the period of revolution of planet A, \( T_A \), is 8 times that of planet B, \( T_B \), we have \( T_A = 8T_B \). Substitute this into the proportion: \( \left(\frac{8T_B}{T_B}\right)^2 = \frac{R_A^3}{R_B^3} \).

Step 3: Simplifying the Equation

Simplify the equation: \( 8^2 = \frac{R_A^3}{R_B^3} \). This gives us \( 64 = \frac{R_A^3}{R_B^3} \).

Step 4: Solving for the Distance Ratio

To find \( \frac{R_A}{R_B} \), take the cube root of both sides: \( \frac{R_A}{R_B} = \sqrt[3]{64} \). Calculating the cube root gives \( \frac{R_A}{R_B} = 4 \).

Key Concepts

Planetary Motion

Planetary motion refers to the movement of planets around a star, like our solar system's planets revolving around the Sun. There's a fascinating law that governs this motion named after Johannes Kepler. He observed that planets follow elliptical orbits and deduced three fundamental laws to describe their motion. Let's focus on the third law of Kepler, which is crucial for understanding the mechanics of how planets orbit.
Kepler's Third Law states that the square of a planet's orbital period (the time it takes for a planet to complete one full orbit around the sun) is directly proportional to the cube of the semi-major axis of its orbit. The semi-major axis is the longest diameter of an elliptical orbit.
This can be mathematically expressed as:

• \(T^2 \propto R^3\)

where:

• \(T\) is the period of revolution.
• \(R\) is the semi-major axis, or distance from the sun.

Understanding this relationship helps us predict how changes in a planet's orbit affect its orbital characteristics, like how much time it takes to go around the sun.

Period of Revolution

The period of revolution is the time a planet takes to make one complete orbit around the sun. For Earth, this period is about 365.25 days, which we define as a year. Different planets have different periods due to their varying distances from the sun. Kepler's Third Law helps in calculating these periods when the distance is known.
The law indicates that if you know the distance of a planet from the sun, you can compute how long it takes for that planet to orbit the sun. The period is crucial because it helps astronomers understand the orbital mechanics and gravitational influences of celestial bodies.
If we apply this to the original exercise, we learn that the period of planet A is 8 times that of planet B. Using Kepler's Third Law, this directly affects the distance each of these planets is from the sun, which can be quantified by further calculations.

Distance From the Sun

Distance from the sun is a vital factor that influences the period of revolution according to Kepler's Third Law. The distance determines the gravitational pull the sun exerts on the planet, affecting how quickly or slowly it orbits. The greater the distance (semi-major axis length), the longer the orbit takes to complete.
In the context of the exercise, determining how much greater the distance of planet A is from the sun compared to planet B involves understanding the proportionality described by Kepler's law:

• If planet A has a period of revolution 8 times that of planet B, this translates via Kepler's Third Law into a calculation challenge.
• The equation from the solution \(T_A = 8T_B\) squared, results in \((8T_B)^2 = T_A^2\).
• Through algebra, this leads us to \(64 = \frac{R_A^3}{R_B^3}\), meaning \(R_A = 4R_B\).

This finding shows that the distance of planet A is exactly 4 times greater than that of planet B. By understanding these principles, students can solve similar problems involving planetary distances and periods with greater confidence.