Question

Given below are two statements: Statement I: A satellite is moving around earth in an orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II: The time period of revolution of the satellite is \[ T = 2\pi \sqrt{\frac{R_e}{g}} \] (for satellite very close to the earth surface), where \(R_e\) is the radius of earth and \(g\) is acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below.

• A. Statement I is true but Statement II is false
• B. Both Statement I and Statement II are true
• C. Statement I is false but Statement II is true
• D. Both Statement I and Statement II are false

Hint:

For satellites near the earth surface, the time period depends only on earth’s radius and acceleration due to gravity.

Answer 1

The Correct Option is D

The question involves evaluating two statements related to the time period of revolution of a satellite moving in an orbit very close to the Earth's surface. 

Statement I: A satellite is moving around Earth in an orbit very close to the Earth's surface. The time period of revolution of the satellite depends upon the density of Earth.

Statement II: The time period of revolution of the satellite is \(T = 2\pi \sqrt{\frac{R_e}{g}}\)(for a satellite very close to the Earth's surface), where \(R_e\) is the radius of Earth and \(g\) is acceleration due to gravity.

Let's analyze each statement:

1. Analysis of Statement I:
   - The time period of revolution of a satellite close to the Earth's surface is determined by the gravitational force, which is influenced by the mass and radius of Earth.
   - The mass of Earth can be expressed in terms of its density (\(\rho\)) and volume (\(V\)).
   - Volume \(V\) of Earth is proportional to \(R_e^3\), so the mass \(M\) is proportional to the density \( \rho \times R_e^3\).
2. Analysis of Statement II:
   - The expression given for the time period of revolution is \(T = 2\pi \sqrt{\frac{R_e}{g}}\).
   - In reality, the correct formula for the time period of a satellite very close to the Earth's surface is derived from Kepler’s third law and is expressed as \(T = \sqrt{\frac{4\pi^2 R_e}{g}}\).
   - Therefore, Statement II is mathematically incorrect.

Conclusion:

• Based on the above analysis, both Statement I and Statement II are false. Thus, the correct answer is: Both Statement I and Statement II are false.