Gravity of the Sun and the Earth – Explanation

Key Concepts

• The motion of the planets around the Sun
• The motion of the Moon around the Earth
• The motion of the satellite around the Earth

Introduction:

There is Gravitational force everywhere in space. In space, there is less gravity, but it is not zero. Gravitation force acts even in a vacuum. Astronauts in space are always subjected to the gravitational force of Earth, other planets, and the Sun. The gravitational pull binds all celestial objects together and causes them to move along a fixed trajectory.

Explanation:

Planets move around the Sun in circular orbits. The necessary centripetal force is provided by the gravitational force exerted by the Sun.

The Centripetal force (𝐹𝑐) is contributed by the gravitational force of attraction (𝐹𝑔) exerted on the Earth by the Sun.

Mass of the Sun = M

Mass of the Earth = m

The centripetal acceleration of the Earth = a_c

F_g = F_c  = ma_c

GMm/r² = mv²/r

Hence the orbital velocity v for the Earth is given by

V_earth= √GM/r

Here G = Universal Gravitational Constant

The Moon revolves around the Earth in a circular orbit the necessary centripetal force is provided by the gravitational force exerted by the Earth.

The centripetal force (𝐹𝑐) is contributed by the gravitational force of attraction (𝐹𝑔) exerted on the Moon by the Earth.

The centripetal acceleration of the Moon = a_c

F_g = F_c  = ma_c    

GMm/r² = mv²

Hence the orbital velocity v for the Moon is given by:

V_moon = √GM/r

Here G = Universal Gravitational Constant

• If we increase the mass of the moon the gravitational force increases, but the orbit remains circular.

• If we increase the mass of the Earth (planet) the Moon revolves around the Earth (planet) in an elliptical orbit.

When a Satellite revolves around the Earth in a Circular Orbit the necessary Centripetal force (𝐹𝑐) is contributed by the gravitational force of attraction (𝐹𝑔) exerted on the satellite by the earth. As there is no displacement in the direction of force work done by the gravitational force on the Satellite in a circular path is zero.

The Centripetal force (𝐹𝑐) is contributed by the gravitational force of attraction (𝐹𝑔) exerted on the satellite by the earth.

Mass of the Earth = M

Mass of the Satellite = m

The centripetal acceleration of the Satellite = a_c

F_g = F_c  = ma_c    

GMm/r² = mv²

Hence the tangential Velocity v is given by

V_ob = √GM/r

Here G = Universal Gravitational Constant

Weightlessness inside a satellite:

Bodies have the weight due to the force of gravity exerted by the Earth on them. In a satellite, the downward force of gravity is balanced by the upward centrifugal force. This balance of force of gravity makes bodies weightless in satellite. Centrifugal force is a pseudo force acting along the radius away from the center.

Bodies have the weight due to the force of gravity exerted by Earth on them. In a satellite, the downward force of gravity is balanced by the upward centrifugal force. This balance of force of gravity makes bodies weightless in satellite. Thus the bodies in a satellite are in a freefall situation.

The launch vehicle first rises initially vertically. The launch vehicle takes the satellite above the Earth’s atmosphere and places it to move tangentially (i. e. at right angles to the line connecting the satellite and the Earth’s center). The speed (V) required depends on the altitude.

Orbital Velocity = V_ob = √GM/r

• Orbital velocity doesn’t depend on the mass of the satellite.
• If we increase the mass of the planet the orbital velocity also increases.
• In absence of gravity, the satellite moves tangentially out of the gravitational field of the earth.

• If speed v < orbital velocity, the spacecraft will descend to Earth in a (decaying) elliptical orbit.
• If speed v > orbital velocity, the spacecraft will ascend into a large elliptical orbit.
• If speed is v >> orbital velocity, the Spacecraft will escape the Earth’s gravity on a parabolic orbit.

Circular and elliptical paths provide bound orbits to a satellite. An object with a velocity equal to the orbital velocity can be placed in these orbits. Whereas the objects with higher orbital speed follow a parabolic or hyperbolic path and escape in unbound orbits.

Question 1

What will happen if an orbiting satellite comes to stand still suddenly?

a. The satellite will move along the tangent.

b. The satellite will move radially towards the center of the earth.

c. The satellite will go to outer space and will be lost.

d. The satellite will continue to move in the same orbit.

Answer: a. The satellite will move along the tangent.

Question 2

An astronaut orbiting the earth in a circular orbit around the earth gently drops a spoon out of the spaceship. It will:

a. Fall vertically down to earth

b. Move towards the moon

c. Move in an irregular way and then fall down to earth

d. Move along with the spaceship

Answer: d Move along with the spaceship.

Question 3

A satellite in a circular orbit around the Earth moves at a constant speed. This orbit is maintained by the force of gravity between the Earth and the satellite. Yet no work is done on the satellite. How is it possible?

a. No work is done if there is no contact between objects.

b. No work is done because there is no gravity in space.

c. No work is done if the direction of motion is perpendicular to the force.

d. No work is done if objects move in a circle.

Answer: c. No work is done if the direction of motion is perpendicular to the force.

Summary

• Planets move around the Sun in circular orbits. The necessary centripetal force is provided by the gravitational force exerted by the Sun. Similarly, the Moon revolves around the Earth in a circular orbit the necessary centripetal force is provided by the gravitational force exerted by the Earth.
• The centripetal force ( ) is contributed by the gravitational force of attraction ( ) then
• Mass of the large body at the center of the circular path = M
• Mass of the revolving body = m
• The centripetal acceleration of the revolving body = a_c

F_g = F_c  = ma_c

GMm/r² = mv²

• Hence the orbital velocity v is given by

V_ob = √GM/r

• Here G = Universal Gravitational Constant.

Thus, orbital velocity = V_ob = √GM/r then,

• If speed v < orbital velocity, the spacecraft will descend to Earth in a (decaying) elliptical orbit.
• If speed v > orbital velocity, the spacecraft will ascend into a large elliptical orbit.
• If speed is v >> orbital velocity, the Spacecraft will escape earth’s gravity on parabolic orbit.