Problem 75 - Master Resource Book in JEE Main Physics

Question

Out of the following, the only correct statement about satellites is (a) A satellite cannot move in a stable orbit in a plane passing through the earth's centre (b) Geostationary satellites are launched in the equatorial plane (c) We can use just one geostationary satellite for global communication around the globe (d) The speed of satellite increases with an increase in the radius of its orbit

Step-by-Step Solution

Answer

Statement (b) is correct: Geostationary satellites are launched in the equatorial plane.

Step 1: Understanding the Context

To solve the problem, we need to evaluate each statement and determine its correctness based on our understanding of satellites and their orbits around Earth.

Step 2: Evaluating Statement (a)

Statement (a) suggests that a satellite cannot move in a stable orbit in a plane passing through the Earth's center. This is incorrect because all stable orbits must pass through the Earth's center due to the gravitational force acting towards this point.

Step 3: Evaluating Statement (b)

Statement (b) indicates that geostationary satellites are launched in the equatorial plane. This is true, as geostationary satellites orbit in the Earth's equatorial plane at a position where their orbital period matches the Earth's rotation period, allowing them to remain stationary relative to a point on Earth.

Step 4: Evaluating Statement (c)

Statement (c) asserts that one geostationary satellite can provide global coverage. This is false because a single geostationary satellite will only cover approximately one-third of the Earth's surface, requiring several satellites for full global coverage.

Step 5: Evaluating Statement (d)

Statement (d) suggests the speed of a satellite increases with an increase in the radius of its orbit. This is incorrect; as the radius of the orbit increases, the gravitational pull weakens, resulting in a decrease in the satellite's speed.

Key Concepts

Geostationary Satellites

Geostationary satellites play a crucial role in communication and meteorology by providing a stable point of reference in the sky. These satellites orbit the Earth synchronously with the planet's rotation. This means a geostationary satellite orbits at the same angular velocity as the Earth, completing one orbit every 24 hours.

To maintain this match, these satellites are placed in an orbit at about 35,786 kilometers above the Earth's equator. This height ensures they remain stationary relative to the Earth's surface. By positioning the satellites in such an orbit, they continuously cover a specific area, making them ideal for applications like broadcasting and weather monitoring.

It's important to note that no single geostationary satellite can provide global coverage. Since each satellite covers only about one-third of the Earth, several satellites are needed to ensure a global service network.

Orbital Mechanics

Orbital mechanics is a fascinating branch of physics that explores how objects move in space under the influence of gravitational forces. When it comes to satellites, the key is the balance between the gravitational pull of the Earth and the satellite's inertia.

For a satellite to maintain a stable orbit, it must travel at a specific speed depending on its distance from the Earth. In a geostationary orbit, this precise speed keeps the satellite synchronized with the Earth’s rotation. The further the satellite is from the Earth, the slower it has to travel to remain in orbit. This principle is in line with Kepler’s third law of planetary motion.

When satellites travel closer to Earth, they experience a stronger gravitational force, necessitating a faster speed to maintain orbit. Conversely, in higher orbits, where gravitational forces are weaker, satellites require slower speeds.

Earth's Gravitational Force

Earth's gravitational force acts as a central pull, keeping satellites in a stable orbit around the planet. This force stems from the mass of the Earth and diminishes with distance. The force can be calculated using the formula: \[ F = \frac{{G \cdot m_1 \cdot m_2}}{{r^2}} \] where:

• \( F \) is the gravitational force,
• \( G \) is the gravitational constant,
• \( m_1 \) and \( m_2 \) are the masses of the Earth and the satellite,
• \( r \) is the distance between the centers of Earth and the satellite.

As a satellite moves further away from Earth, \( r \) increases, causing \( F \) to decrease. This diminishing force means that the satellite doesn’t need to travel as fast to maintain its orbit. This relationship explains why higher altitude satellites, like those in geostationary orbits, have slower orbital speeds compared to low Earth orbit satellites.