GRAVITATIONAL FIELD: learn Escape velocity, gravitational potential, satellite, parking orbit, Newton's law of gravitation

We have discussed Electric Field and Magnetic field. We want to know what Gravitational Fields is all about.

Gravitational field is a region where gravitational force is experienced. According to Newton, he said there is always an attractive force between two bodies. Let's look at Newton's law of gravitation

This law state that the gravitational force between two bodies of mass m1 and mass m2 is directly proportional to their product and inversely proportional to the square of their distance apart.i.e 

Where m1 and  m2 are the masses of the bodies in kilogram, G is the gravitational constant in Nm²kg⁻² , r is the distance between them in metres and F is the gravitational force.

What can we say from that formula, The gravitational force(F) is inversely proportional to the square of the distance i.e the greater the distance between two bodies, the smaller the Force and vice versa.

Let's take an example on this

• Calculate the gravitational force of attraction between two bodies of masses 2×10¹³kg and 3×10¹²kg separated by a distance of 4m(G= 6.67×10⁻¹¹Nm²kg⁻²)

Solution

m1=2×10¹³kg, m2= 3×10¹²kg, r=4m, 

Unlike electric force and magnetic force, where we have force of attraction and repulsion, gravitational forces are always those of attraction. It is this attraction that keeps the moon in its orbit around the earth and also the earth and the other planets in their orbits round the sun. The earth attracts all bodies towards its centre. When you throw a body up, the body will get to a certain height and then fall back towards the centre of the earth.

Another formula for gravitational force can be derived between the earth and a body.

Where me is the mass of the earth, m is the mass of the body. Let's take a question on this.

• The mass of the earth is 6.0×10²⁴kg and that of the moon is 7.0×10²²kg. If the distance between them is 4×10⁸m. Calculate the force of attraction between them (G= 6.67×10⁻¹¹Nm²kg⁻²)

Solution

me=6.0×10²⁴kg , m=7.0×10²²kg, r= 4×10⁸m

Let's look at the relationship between acceleration due to gravity(g) and gravitational constant(G).

From the first formula with asterisk, it can be seen that acceleration due to gravity(g) is inversely proportional to the square of the distance(r), which means that the higher you go, the less the value of (g).

 Let's take an example on this

• If the radius of the earth is 6.38×10⁶m. What is the mass of the earth?(G=6.67×10⁻¹¹Nm²kg⁻², g= 9.8m/s²)

Solution

Gravitational Potential

Gravitational potential is how much a brick can hurt your toe when dropped from a height. The higher the brick when dropped, the more your toe will hurt…! We say that the brick, when held at a height, has the potential to hurt your toe, because of gravity.

Gravitational potential is the workdone in moving unit mass from infinity to the point of consideration.

The formula for finding gravitational potential is

The negative sign indicate that work is done in moving a body away from another. Negative here is indication and not the real number line’s integer section.

Escape velocity

The earth gravitational force attracts bodies toward its centre, for a body not to be attracted to the earth by its gravitational force, that body needs to be outside it's region. The minimum velocity required by a body to escape from the earth gravitational field is called ESCAPE VELOCITY. It can be calculated by 

Where R is the radius of the earth

Satellite

An orbit is a regular repeating path that an object in space takes around another object. An object in an orbit is called a satellite. A Satellite can be natural(moon) or man-made.

To put a man-made satellite in orbit, it must enter the orbit at the correct speed where the centripetal force needed to keep it in the orbit is equal to the force of gravity on it.i.e

Where v is the velocity of satellite in orbit, r is the radius of the earth and g is acceleration due to gravity

A satellite is in a parking orbit if its period is equal to the period of the earth.

Question

• A satelite moves in a circular orbit of radius 4R round the earth. The acceleration of the satelite in terms of g is

Solution

Exercises

1. A satellite moves in a circular orbit of radius 2R round the earth. What is the acceleration of the satelite?
2. Two bodies of the same mass experience a force of attraction 2.82 × 10⁻⁷N when placed at a distance of 100cm in a gravitational field. Determine their mass [G = 6.67 × 10⁻¹¹Nm²Kg⁻²]
3. If the radius of the earth is 6.4 x 10⁶m, the escape velocity of a satellite from the earth is
4. A man and his friend have masses 70kg and 60kg respectively. If they are seated 1.0m apart. Calculate the gravitational force of attraction between them (6.67 × 10⁻¹¹Nm²Kg⁻²)

Answer the questions and write your answers in the comment section below