Question

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

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

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.

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.

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.

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.

Key concepts

Artificial Satellites

Artificial satellites are human-made objects that orbit the Earth or other celestial bodies. These satellites have a wide range of applications, from communication to weather forecasting, and scientific research. Understanding how they function involves a variety of principles of physics and astronomy. One crucial factor for an artificial satellite to remain in orbit is its speed.

• They must maintain a certain orbital velocity to stay in orbit without drifting away or falling back to Earth.
• They are launched into space by rockets and placed into orbit at specific distances from Earth, depending on their intended purpose.

An example of a satellite is a weather satellite that helps in predicting weather conditions by taking pictures of cloud formations and other meteorological data. This broad utility makes artificial satellites invaluable in today's world.

Gravitational Constant

The gravitational constant, denoted as \( G \), plays a crucial role in the calculation of the forces involved in planetary motion and the functioning of satellites. This constant is a measure of the strength of gravity, the force of attraction between two bodies. Isaac Newton discovered it at the end of the 17th century. It is defined as the force between two one-kilogram masses that are one meter apart.

• The value of \( G \) is approximately \( 6.674 imes 10^{-11} \, \text{Nm}^2/\text{kg}^2 \).
• It facilitates the calculation of gravitational forces using Newton's law of universal gravitation.
• In satellite mechanics, \( G \) is used to calculate the orbital velocity and the forces acting on the satellite.

Understanding this concept is key to comprehending how and why satellites move the way they do.

Orbital Speed

Orbital speed is the velocity at which a satellite travels to maintain a stable orbit around a celestial body. It is crucial for the satellite not to fall back to Earth or veer off into space. The orbital speed depends on the gravitational pull of the Earth and the distance from the center of the Earth to the satellite.

Key Factors Influencing Orbital Speed:

• The mass of the celestial body being orbited. Larger masses exert greater gravitational force, affecting orbital speed.
• The distance from the satellite to the center of the planet. As this distance increases, the required speed decreases.
• Using the formula: \[ v = \sqrt{\frac{GM}{r}} \], where:
  • \( v \) is the orbital velocity
  • \( G \) is the gravitational constant
  • \( M \) is the mass of the Earth
  • \( r \) is the distance from the Earth's center to the satellite

For a satellite to orbit close to Earth, a speed of roughly \( 8 \, \text{kms}^{-1} \) is necessary. This speed ensures the satellite balances between falling towards Earth due to gravity and moving forward in its orbital path.

Geostationary Satellite

Geostationary satellites are a special type of artificial satellite with a unique orbiting characteristic. These satellites revolve around Earth at the same rate that Earth rotates, making them appear stationary relative to the surface. As a result, they maintain a fixed position above the equator at approximately 36,000 kilometers above Earth.

• Their period of revolution matches Earth’s rotation period, which is about 24 hours.
• This unique trait makes them particularly useful for communication and weather observation because they offer consistent coverage over specific areas.
• To achieve this geostationary orbit, satellites must be positioned precisely and maintain the correct speed and altitude.

Geostationary satellites are vital in the modern world for facilitating international communication and providing dependable weather data, contributing significantly to global meteorological monitoring and forecasting.