Problem 76 - Master Resource Book in JEE Main Physics

Question

A satellite \(S\) is moving in an elliptical orbit around earth. The mass of the satellite is very small compared to the mass of the earth? (a) The acceleration of \(S\) is always directed towards the centre of the earth (b) The angular momentum of \(S\) about the centre of the earth changes in direction but its magnitude remains constant (c) The total mechanical energy of \(S\) varies periodically with time (d) The linear momentum of \(S\) remains constant in magnitude

Step-by-Step Solution

Answer

Statements (a) and (b) are true; (c) and (d) are false.

Step 1: Understanding the problem

We are given a satellite moving in an elliptical orbit around the Earth. We need to determine which statements about the satellite are true based on known principles of physics.

Step 1: Acceleration direction

For any object in elliptical motion under the influence of gravity, the gravitational force (and hence the acceleration) is always directed towards the center of the Earth. Thus, statement (a) is true.

Step 2: Angular momentum

Angular momentum of the satellite about the center of the Earth is conserved because there is no external torque acting on the satellite. Thus, while the direction of the angular momentum vector might change as it orbits, its magnitude remains constant. Hence, statement (b) is true.

Step 3: Total mechanical energy

In an elliptical orbit, the total mechanical energy, which is the sum of kinetic and potential energy, remains constant over time as there is no external force doing work on the satellite. Thus, statement (c) is false.

Step 4: Linear momentum

The linear momentum of the satellite does not remain constant in magnitude due to the changing speed as the satellite moves through its elliptical orbit. Therefore, statement (d) is false.

Key Concepts

Elliptical Orbit

An elliptical orbit is a path that takes the shape of an elongated circle, defined mathematically as an ellipse. Satellites that orbit in such a path around a planet like Earth are not moving in circles, but in more stretched curves. The Earth is located at one of the two focal points of the ellipse.
This means the satellite's distance from Earth changes as it moves along its orbit. When the satellite is closer to Earth (at the perigee), it moves faster due to the stronger gravitational pull. Conversely, when it is further away (at the apogee), it slows down. This variation in speed is what characterizes an elliptical orbit.
Understanding elliptical orbits is crucial in satellite motion as it explains why satellites have varying speeds and distances to the Earth during their orbit.

Gravitational Force

Gravitational force is the attractive force that exists between any two masses. For a satellite orbiting Earth, this force is what keeps it in orbit and prevents it from flying off into space.
This force acts along the line joining the center of the Earth and the satellite. Therefore, no matter where the satellite is in its elliptical orbit, the gravitational pull is always directed towards the center of the Earth. This is why the satellite's acceleration is directed inwards, maintaining its orbit over time.
The strength of this force depends on two main factors: the masses of the objects involved (such as the Earth and the satellite) and the distance between them. This principle is governed by Newton's law of universal gravitation.

Angular Momentum

Angular momentum is a measure of the rotational motion of a body and how its mass is distributed in relation to the axis of rotation. For a satellite in orbit, angular momentum relative to Earth's center is conserved due to the absence of external torques.
This means that even though the satellite's direction in space might change as it follows its elliptical path, the overall amount or magnitude of its angular momentum remains constant.
The conservation of angular momentum is a fundamental principle in physics, indicating that in an isolated system (like a satellite orbiting Earth without external torque), the total angular momentum does not change, preserving the satellite's path and speed.

Mechanical Energy

Mechanical energy in the context of a satellite orbiting Earth is the sum of its kinetic energy (due to its speed) and potential energy (due to its position in Earth's gravitational field).
Although the satellite's speed and height above Earth vary throughout its elliptical orbit, the total mechanical energy remains constant if no external forces act on it. This constancy is because the decrease in potential energy when the satellite moves closer to Earth is balanced by an increase in kinetic energy, and vice versa.
Understanding mechanical energy is significant for analyzing satellite motion without external influences, affirming its stability and predictability in its orbit.