Problem 71 - Master Resource Book in JEE Main Physics

Question

A satellite orbits the earth at a height of \(400 \mathrm{~km}\) above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite \(=200 \mathrm{~kg}\), mass of the earth \(=6.0 \times 10^{24} \mathrm{~kg}\), radius of the earth \(=6.4 \times 10^{6} \mathrm{~m}, G=6.67 \times 10^{-11} \mathrm{~N}-\mathrm{m}^{2} / \mathrm{kg}^{2}\). (a) \(5.2 \times 10^{10} \mathrm{~J}\) (b) \(3 \times 10^{6} \mathrm{~J}\) (c) \(4 \times 10^{6} \mathrm{~J}\) (d) \(6 \times 10^{9} \mathrm{~J}\)

Step-by-Step Solution

Answer

Energy required is approximately \(5.2 \times 10^{10} \, \text{J}\), so option (a) is correct.

Step 1: Understanding Gravitational Potential Energy

The energy required to remove the satellite from Earth's gravitational pull is equal to the gravitational potential energy at its orbit. This can be calculated using the formula: \[ U = - \frac{G M m}{r} \]where \( U \) is the gravitational potential energy, \( G \) is the gravitational constant \( 6.67 \times 10^{-11} \, \text{N}-\text{m}^2/\text{kg}^2 \), \( M \) is the mass of the Earth \( 6.0 \times 10^{24} \, \text{kg} \), \( m \) is the mass of the satellite \( 200 \, \text{kg} \), and \( r \) is the distance from the center of the Earth to the satellite, which is the sum of the Earth's radius \( 6.4 \times 10^6 \, \text{m} \) and the satellite's height above the Earth \( 400 \, \text{km} = 400,000 \, \text{m} \). Therefore, \( r = 6.8 \times 10^6 \, \text{m} \).

Key Concepts

Satellite Orbit

A satellite orbit refers to the trajectory a satellite follows around a celestial body, such as Earth, due to gravity. Understanding orbits is crucial because they determine the altitude, velocity, and duration for which a satellite remains in space without crashing back to Earth or flying away into infinitude.

• **Types of Orbits:** Satellites can be in low Earth orbit (LEO), medium Earth orbit (MEO), or geostationary orbit (GEO), each serving different purposes.
• **Orbital Velocity:** The speed needed for a satellite to stay in orbit is known as orbital velocity. It balances gravitational forces and inertia.
• **Orbital Period:** This is the time a satellite takes to complete one full orbit around Earth.

In the given problem, the satellite orbits 400 km above Earth's surface. The radius of this orbit is found by adding Earth's radius to the altitude of the satellite, creating a balance between gravitational pull and orbital velocity to maintain its path.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a crucial part of gravitational calculations. This universal constant is used in Newton's law of universal gravitation to describe the attractive force between two bodies.

• **Value of G:** The gravitational constant is approximately \( 6.67 \times 10^{-11} \, \text{N}-\text{m}^2/\text{kg}^2 \).
• **Role:** It helps calculate the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \).
• **Application:** In our exercise, it's used to find gravitational potential energy, which measures the energy of a system consisting of Earth and the satellite.

By applying the formula \( F = \frac{G M m}{r^2} \), where \( M \) and \( m \) are the masses of Earth and the satellite, respectively, we understand how gravitational attraction governs the motion and energy interactions in space.

Earth's Gravitational Pull

Earth's gravitational pull is the force exerted by Earth's mass on objects within its vicinity. This pull keeps satellites, like the one in our exercise, in orbit and dictates how much energy is required to move them.

• **Formula:** The energy required to escape Earth’s pull from a given orbit is the gravitational potential energy, given by \( U = - \frac{G M m}{r} \).
• **Gravitational Influence:** At the satellite’s orbital altitude, Earth's gravity is slightly weaker than on the surface but still substantial enough to maintain an orbit.
• **Escape Energy:** A satellite needs a specific amount of energy to overcome this pull, measured by calculating the change in gravitational potential energy from the orbit to a theoretical point infinitely far away.

This gravitational force is essential for the functioning of satellites, ensuring they remain in orbit and must be understood when planning missions to either launch or deorbit satellites effectively.