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Chapter 6: Problem 775

Which one of following statements regarding artificial satellite of earth is incorrect (A) The orbital velocity depends on the mass of the satellite (B) A minimum velocity of \(8 \mathrm{kms}^{-1}\) is required by a satellite to orbit quite close to the earth. (C) The period of revolution is large if the radius of its orbit is large (D) The height of geostationary satellite is about \(36000 \mathrm{~km}\) from earth

Short Answer

Expert verified
The incorrect statement is (A). The orbital velocity of a satellite depends on the mass of the planet and the distance between the center of the satellite and the center of the planet, not on the mass of the satellite.

Step by step solution

01

Statement (A)

The orbital velocity depends on the mass of the satellite. This statement is incorrect. The orbital velocity of a satellite depends on the mass of the planet (or celestial body) it is orbiting and the distance between the center of the satellite and the center of the planet. It does not depend on the mass of the satellite. The formula for the orbital velocity is \(v = \sqrt{\frac{GM}{r}}\), where \(v\) is the orbital velocity, \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(r\) is the distance between the center of the satellite and the center of the planet.
02

Statement (B)

A minimum velocity of \(8 \mathrm{kms}^{-1}\) is required by a satellite to orbit quite close to the Earth. This statement is correct. To have a stable orbit close to the Earth, a satellite must have a minimum orbital velocity of around \(8 \mathrm{kms}^{-1}\). This is called the "orbital speed" or "circular speed," which keeps the satellite from falling back to Earth.
03

Statement (C)

The period of revolution is large if the radius of its orbit is large. This statement is correct. The period of revolution of a satellite is the time it takes for the satellite to complete one orbit around the Earth. It can be calculated using the formula \(T = 2\pi \sqrt{\frac{a^3}{GM}}\), where \(T\) is the period of revolution, \(a\) is the semi-major axis of the orbit, and \(G\) and \(M\) are the gravitational constant and the mass of the planet, respectively. As the radius of the orbit increases, so does the period of revolution.
04

Statement (D)

The height of a geostationary satellite is about \(36000 \mathrm{~km}\) from Earth. This statement is correct. A geostationary satellite is a satellite that remains fixed above a specific point on Earth's surface, and its period of revolution is equal to Earth's period of rotation (24 hours). This can be achieved by placing the satellite in a circular, equatorial orbit with an altitude of about \(36000 \mathrm{~km}\) from Earth. So, the incorrect statement is (A), as the orbital velocity does not depend on the mass of the satellite but rather on the mass of the planet and the distance between the satellite and the planet.

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