A satellite S is moving in an elliptical orbit around the 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)Th e lin ear momen tum of S remains constant in magnitude.Correct answer is option 'A,C'. Can you explain this answer?

Question

Answer

A) The acceleration of S is always directed towards the centre of the earth:
The acceleration of an object moving in a circular or elliptical orbit is always directed towards the center of the orbit. This is because the force of gravity acts as the centripetal force that keeps the satellite in its orbit. In the case of a satellite orbiting the Earth, the gravitational force between the Earth and the satellite provides the necessary centripetal force.

The acceleration of the satellite is given by the equation:
a = GM/r^2
where G is the gravitational constant, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite. As the satellite moves in its elliptical orbit, the distance r changes, but the direction of the acceleration always points towards the center of the Earth.

B) The angular momentum of S about the center of the earth changes in direction, but its magnitude remains constant:
Angular momentum is a property of rotating objects and is defined as the product of the moment of inertia and the angular velocity. In the case of the satellite, its angular momentum is given by:
L = Iω
where I is the moment of inertia and ω is the angular velocity.

In an elliptical orbit, the satellite's angular velocity changes as it moves closer or farther from the Earth. However, the satellite's moment of inertia remains constant, as it is determined by the mass and distribution of mass of the satellite itself, which does not change.

Therefore, although the direction of the angular momentum changes as the satellite moves in its elliptical orbit, the magnitude of the angular momentum remains constant.

C) The total mechanical energy of S varies periodically with time:
The total mechanical energy of an object is the sum of its kinetic energy and potential energy. In the case of the satellite, its mechanical energy is given by:
E = K + U
where K is the kinetic energy and U is the potential energy.

In an elliptical orbit, the distance between the satellite and the Earth varies, resulting in changes in both the kinetic and potential energy. As the satellite moves closer to the Earth, its kinetic energy increases while its potential energy decreases, and vice versa when it moves farther from the Earth.

Therefore, the total mechanical energy of the satellite varies periodically with time as it moves in its elliptical orbit.

D) The linear momentum of S remains constant in magnitude:
Linear momentum is defined as the product of an object's mass and its velocity. In the case of the satellite, its linear momentum is given by:
p = mv
where m is the mass of the satellite and v is its velocity.

In an elliptical orbit, the satellite's velocity changes as it moves closer or farther from the Earth. However, the mass of the satellite remains constant.

Therefore, although the velocity of the satellite changes, the mass remains constant, resulting in the linear momentum of the satellite also remaining constant in magnitude.

Top Courses for JEEView all

Physics for JEE Main & Advanced

Mock Tests for JEE Main and Advanced 2025

Chemistry for JEE Main & Advanced

Mathematics (Maths) for JEE Main & Advanced

Chapter-wise Tests for JEE Main & Advanced

Top Courses for JEE

Top Courses for JEE

Suggested Free Tests

Related Content

JEE Courses

Mock Test Series

Past Year Papers

Additional Courses

Company