Gravitational Force and Motion within the Solar System

Introduction:

The gravitational pull binds all celestial objects together and causes them to move along a fixed trajectory.

Nature of Gravitational Force:

The objects move towards the Earth when they are released from a certain height above the ground because the Earth pulls them towards its center.

The force with which the Earth pulls the objects towards itself is called the Earth’s gravitational force or simply gravity.

Though the Earth is not in contact with the object, it can still apply a pulling force on it. A force that can be applied even when two objects are not in contact with each other is called a non-contact force.

Therefore, the gravitational force is a non-contact force.

Some examples where the gravitational force plays a major role are shown in the below image.

Inverse-square Law:

In the year 1666 Newton studied Kepler’s work on planetary motion. He found that the magnitude of the force F, on a planet due to the Sun varies inversely with the square of the distance r between the center of the planet and the Sun.

F α 1/r²

This force acts in the direction of the line connecting the centers of the two objects.

Proof:

The gravitational force follows the inverse square law. For example, if the weight of an apple on the surface of the Earth (radius = d) is 1N as we increase the distance, such as 2d, 3d, 4d, and 5d, the weight of the apple becomes (1/4)N, (1/9)N and (1/16)N.

Example:

Similarly, suppose the force between two bodies is 80N when they are at a distance of 0.1m. If we keep increasing the distance between the two masses as shown in the table, we get a graph between force and distance as shown in the figure.

According to Newton’s third law, we know that every action has an equal and opposite reaction.

Direct Relationship:

Based on his third law of motion and experimental findings, Newton concluded that the gravitational force of attraction between two objects must be proportional to the product of their masses.

If A and B are two bodies with masses m₁ and m₂, separated by a distance r. The force F between two objects is given by,

F α m₁ x m₂

Thus, the gravitation force acting between two bodies depends on the product of the mass of the bodies and the distance between the two bodies. If we increase the masses, the force increases and if we increase the distance between the bodies, the gravitational force decreases.

Weight of a Body:

The Earth attracts everybody towards its center with a certain force that depends on the mass of the body and the gravitational acceleration at that place. Weight is determined by the force with which a body is attracted toward the center of the Earth.

Weight of a body on Earth = Gravitational force exerted by the Earth on the body

W = mass of the body x gravitational acceleration

W = mg

Weight is measured using a spring balance.

An Object Weightlessness on the Surface of the Moon:

It has been experienced that when astronauts land on the surface of the Moon, they feel less weight.

If we take an object to the Moon, it will be weightless because the Moon’s gravity is weaker than the Earth’s as the former has lesser mass. However, the mass of the object will remain the same.

Size of Earth = 4 x Size of Moon

Gravitational acceleration on Earth =  g_e

Gravitational acceleration on Moon =

Weight on Earth = mg_e

Weight on Moon = mg_m

Weight on Moon = (1/6) x Weight on Earth

Question:

What would be the weight of a person on the Moon and Earth whose mass is 12kg?

Answer:

Weight of the person on earth

= mg_E  = 12kg x 9.8 m/s² = approximately 120 N

Weight of the person on moon = mg_m = 120N/6 = 20N

The Weight of an Object on the Other Planets:

We know that weight = Mass x Surface gravity

Here surface gravity is the gravitational force exerted on the surface.

We can calculate the weight of an object on the other planets by comparing its weight on the Earth using the concept of relative surface gravity.

Relative means there is a comparison with a reference point, in this case, earth.

We can measure the weight of an object on Earth. Here we are taking the weight on earth as a reference, so the relative surface gravity of the Earth is 1.

Weight of a person on a planet = Mass of the object x Relative surface gravity of the planet

Question:

Susan weighs 180 lbs on Earth. How much would she weigh on Pluto?

Answer:

Mass of Susan = Weight on earth / Relative surface gravity on Earth

Mass of Susan = 180 lbs / 1 = 180 lbs

Weight of Susan on Pluto = Mass of Susan x Relative surface gravity of Pluto

Weight of Susan on Pluto = 180 lbs x 0.06 = 10.8 lbs

Comparing Gravity on Planets:

We can conclude from the graph that on Jupiter the weight of a person will be maximum.

On Pluto, the weight of the person will be minimal.

Weightlessness in Space:

Bodies have 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 satellites.

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 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 satellites. Thus, the bodies in a satellite are in a freefall situation.