Problem 804 - JEE Physics

Question

The earth revolves round the sun in one year. If distance between then becomes double the new period will be years. (A) \(0.5\) (B) \(2 \sqrt{2}\) (C) 4 (D) 8

Step-by-Step Solution

Answer

The new period will be \(2 \sqrt{2}\) years.

Step 1: Write down the given information

The Earth revolves around the Sun in one year, so the period T₁ = 1 year. We need to find the new period, T₂, when the distance between them becomes double.

Step 2: Apply Kepler's Third Law

According to Kepler's Third Law, the square of the period of a planet is directly proportional to the cube of the semi-major axis of its orbit: \(T^2 \propto a^3\) Where T is the period of revolution and a is the semi-major axis.

Step 3: Set up a proportion

Since we're given that the distance between the Earth and the Sun doubles, we can set up a proportion using Kepler's Third Law: \(\frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3}\) We know that \(a_2\) is double that of \(a_1\), so we can substitute \(2a_1\) for \(a_2\): \(\frac{T_1^2}{a_1^3} = \frac{T_2^2}{(2a_1)^3}\)

Step 4: Substitute given values

Now, we can substitute the given values: \(T_1 = 1\) year and \(a_2 = 2a_1\): \(\frac{1^2}{a_1^3} = \frac{T_2^2}{(2a_1)^3}\)

Step 5: Simplify and solve for T₂

To solve for T₂, we can simplify the equation: \(\frac{1}{a_1^3} = \frac{T_2^2}{8a_1^3}\) Now, we can cross multiply: \(8a_1^3 = T_2^2 \cdot a_1^3\) Notice that both sides have the same \(a_1^3\) term, which we can divide to cancel it out: \(8 = T_2^2\) To solve for T₂, we take the square root of both sides: \(T_2 = \sqrt{8}\)

Step 6: Write the final answer

By solving, we find: \(T_2 = 2 \sqrt{2}\) Thus, the new period will be \(2 \sqrt{2}\) years, which corresponds to the option (B).

Key Concepts

Orbital Period

The orbital period is an important concept in astronomy. It's the time a celestial body, such as a planet, takes to complete one full orbit around its star. In our example, the Earth takes one year to orbit the Sun, so its orbital period is one year. The orbital period can change if the orbit's shape or size changes.
A longer or shorter orbital path affects how quickly a planet orbits. When the distance from the star increases, the planet has a longer journey, leading to a longer orbital period. Conversely, if the distance decreases, the orbital period shortens. Understanding orbital periods helps us predict planetary positions and explore different celestial mechanics like calculating space missions. This concept is at the heart of studying any solar system, helping scientists understand how planets move and interact.

Semi-Major Axis

The semi-major axis is a key term when talking about planetary orbits. It represents half of the longest diameter of an elliptical orbit. Orbit paths are often elliptical, shaped like elongated circles, with the sun at one focus. Knowing the semi-major axis helps us calculate the size of the orbit. When the Earth is at its farthest from the Sun, this distance is known as the aphelion and is slightly larger than when it is closest, or at the perihelion. The semi-major axis effectively averages these extremes. It’s crucial in applying Kepler's Third Law as it helps us understand the link between the distance of a planet from its star and the time it takes to complete its orbit.
Often denoted by the letter "a," the semi-major axis’s value plays a significant role in calculating gravitational interactions and energy within orbits. It's a primary factor when determining how changes in orbit shape influence the period.

Proportional Relationship

Proportional relationships are fundamental in understanding Kepler's Third Law. This law states that the square of a planet's orbital period ( T² ) is directly proportional to the cube of the semi-major axis ( a³ ) of its orbit. Simply put, if you know how the semi-major axis changes, you can predict how the orbital period will change as well. This relationship can be expressed as:

• T² ∝ a³
• or T²/a³ = constant.

In practical terms, if the semi-major axis doubles, as in the given example, the orbital period will not just double but will increase to match this proportional change. This is why the calculated result was T₂ = 2√2 years, not simply 2 years. This mathematical relationship helps astronomers comprehend how different factors influence orbit duration, making it valuable for predicting changes in celestial mechanics.