In this article, we shall learn numerical problems to calculate the gravitational force of attraction between two bodies.

Example – 01:

Calculate the gravitational force of attraction between two metal spheres each of mass 90 kg, if the distance between their centres is 40 cm. Given G = 6.67 x 10^-11 N m²/kg². Will the force of attraction be different if the same bodies are taken on the moon, their separation remaining the same?

Given: Mass of first body = m₁ = 90 kg, mass of second body = m₂ = 90 kg, Distance between masses = r = 40 cm = 40 x 10^-2 m, G = 6.67 x 10^-11 N m²/kg² .

To Find: Force of attraction = F =?

Solution:

By Newton’s law of gravitation

If the same bodies are taken on the moon, their separation remaining the same, the force of attraction between the two bodies will remain the same, because the force of attraction between two bodies is unaffected by the presence of the third body and medium between the two bodies.

Ans: The force of attraction between two metal spheres is 3.377 x 10^-6 N

The force of attraction between two bodies remains the same

Example – 02:

Find the gravitational force of attraction between the moon and the earth if the mass of the moon is 1/81 times the mass of earth. G = 6.67 x 10^-11 N m²/kg², radius of moon’s orbit is 3.58 x 10⁵ km. Mass of the earth = 6 x 10²⁴ Kg.

Given: Mass of Moon =  1/81 times the mass of earth, m_m = 1/81 m_e ,Distance between the moon and earth  = r = 3.58 x 10⁵ km = 3.58 x 10⁸ m, G = 6.67 x 10^-11 N m²/kg² . Mass of earth = M_e = 6 x 10²⁴ Kg

To Find: Force of attraction = F =?

Solution:

By Newton’s law of gravitation

Ans: The gravitational force of attraction between the moon and the earth is 2.213 x 10²⁰ N

Example – 03:

Two bodies of masses 5 kg and 6 x 10²⁴ kg are placed with their centres 6.4 x 10⁶ m apart. Calculate the gravitational force of attraction between the two masses. Also, find the initial acceleration of two masses assuming no other forces act on them.

Given: Mass of first body = m₁ = 5 kg, mass of second body = m₂ = 6 x 10²⁴ kg, Distance between masses = r = 6.4 x 10⁶ m, G = 6.67 x 10^-11 N m²/kg² .

To Find: Force of attraction between two masses = F =? Initial accelerations of the two masses =?

Solution:

By Newton’s law of gravitation

Initial acceleration of body of mass 5 kg

By Newton’s second law of motion F = ma

Thus a = F/m = 48.85 / 5 = 9.77 m/s²

Initial acceleration of body of mass 6 x 10²⁴ kg

By Newton’s second law of motion F = ma

Thus a = F/m = 48.85 / 6 x 10²⁴ = 8.142 x 10^-24 m/s²

Ans: The force of attraction between the two masses = 48.85 N

The Initial acceleration of body of mass 5 kg is 9.77 m/s² and

That of body of mass 6 x 10²⁴ kg is 8.142 x 10^-24 m/s².

Example – 04:

A sphere of mass 40 kg is attracted by another spherical mass of 15 kg by a force of 9.8 x 10^-7 N when the distance between their centres is 0.2 m. Find G.

Given: Mass of first body = m₁ = 40 kg, mass of second body = m₂ = 15 kg, force between them = F = 9.8 x 10^-7 N, Distance between the masses = r = 0.2 m.

To Find: Universal gravitation constant = G =?

Solution:

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.533 x 10^-11 N m²/kg².

Example – 05:

A sphere of mass 100 kg is attracted by another spherical mass of 11.75 kg by a force of 19.6 x 10^-7 N when the distance between their centres is 0.2 m. Find G.

Given: Mass of first body = m₁ = 100 kg, mass of second body = m₂ = 11.75 kg, distance between masses = r = 0.2 m, force between them = F =  19.6 x 10^-7 N,

To Find: Universal gravitation constant = G =?

Solution:

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.672 x 10^-11 N m²/kg².

Example – 06:

The distance of a planet from the earth is 2.5 x 10⁷ km and the gravitational force between them is 3.82 x 10¹⁸ N. Mass of the planet and earth are equal, each being 5.98 x 10²⁴ kg. Calculate the universal gravitation constant.

Given: Mass of Planet = m₁ = 5.98 x 10²⁴ kg, mass of earth = m₂ = 5.98 x 10²⁴ kg, distance between them = r = 2.5 x 10⁷ km = 2.5 x 10¹⁰ m, force between them = F =  19.6 x 10^-7 N,

To Find: Universal gravitation constant = G =?

Solution:

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.676 x 10^-11 N m²/kg².

Example – 7:

Three 5 kg masses are kept at the vertices of an equilateral triangle each of side of 0.25 m. Find the resultant gravitational force on any one mass. G = 6.67 x 10^-11 S.I. units.

Given:  m₁ = 5 kg, m₂ = 5 kg, m₃ = 5 kg, r = 0.25 m, G = 6.67 x 10^-11 N m²/kg².

To find: Force on m₁ =?

Solution:

By Newton’s law of gravitation, the force on mass m₁ due to mass m₂.

By Newton’s law of gravitation, the force on mass m₁ due to mass m₃.

The angle between F₁₂ and F₁₃ is 60°. (Angle of an equilateral triangle). The net force on m₁ is given by

The two forces are equal, hence their resultant act along angle bisector towards centroid.

Ans:  Force on any mass is 4.621 x 10^-8 N towards the centroid

Previous Topic: Theory of Newton’s Law of Gravitation

Next Topic: Concept of Gravitational Intensity

Science > Physics > Gravitation > Numerical Problems on Newton’s Law of Gravitation

The post Numerical Problems on Newton’s Law of Gravitation appeared first on The Fact Factor.

In this article, we shall study Newton’s law of gravitation, its universality, and universal gravitation constant.

Newton’s Law of Gravitation:

Statement:

Every particle of matter in the universe attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Explanation:

Let ‘m₁’ and ‘m₂’ be the masses of two particles separated by a distance r as shown. According to Newton’s Law of gravitation, these particles will attract each other by a force ‘F’ such that

Where ‘G’ is a constant of proportionality and known as Universal gravitation constant. The value of ‘G’ in S.I. system is 6.673 10^-11 N m² kg^-2 and in c.g.s. system is 6.673 10^-8 dyne cm² g^-2.

The mathematical expression for the law of gravitation is sometimes written as

The negative sign indicates the force of attraction.

Newton’s Law of Gravitation in Vector Form:

Force of Gravitation:

The force of attraction between two material bodies in the universe is known as the force of gravitation.

If one of the body is the earth or some other planet or natural satellite then the force of gravitation is called the force of gravity.

Characteristics of Gravitational Force:

• The gravitational force between two bodies forms the action-reaction pair. The gravitational force between two masses is always that of attraction. If the first body attracts the second body with force F (direction of force from the second body to the first body), then the second body attracts the first body with equal force F  (direction of force from the first body to the second body).
• The gravitational force between two masses is always acting along the line joining the centre of the two masses. Hence it is a central force.
• The gravitational force between two masses is independent of the medium between the two masses. It means the gravitational force between two masses is the same when they are kept in a vacuum or in water or in the air. This fact rules out the possibility of making gravity screens.
• The gravitational force between two masses is independent of their sizes or distribution of mass of the bodies.
• The gravitational force between two bodies does not depend upon the presence or the absence of other bodies.
• If the masses of the body are small, the gravitational force between them is negligible. If the masses are large like that of the sun and the earth, the gravitational force of attraction is considerable.
• The gravitational force between two bodies is called action – at – a distance type of interaction, because the two particles interact even though they are not in contact with each other. Thus gravitational force is a non-contact force.
• Gravitational force is a conservative force because the work done by the gravitational force is independent of the path between the initial and final position.

The universality of Newton’s Law of Gravitation:

Newton’s law of gravitation is also called as the universal law of gravitation because

• It is applicable to all material bodies irrespective of their sizes. It is applicable to very minute particles like atoms, electrons at the same time it is applicable to heavenly bodies like planets, stars etc.
• The law is applicable to all material bodies irrespective of the distance between them. It is applicable to interatomic distances at the same time it is applicable to stellar distances i.e. the distance between stars.

Evidence Supporting Newton’s Law of Gravitation:

• The Earth moves around the Sun under the gravitational influence of the Sun on the Earth.
• The Moon moves around the Earth under the gravitational influence of the Earth on the Moon.
• The high tide and low tide are caused due to the gravitational influence of the Moon on the Earth.
• The times of lunar eclipses and solar eclipses calculated on the basis of Newton’s law of gravitation are found to be approximately correct.

Difference Between Gravitation and Gravitational Force:

Gravitation is a natural phenomenon by which material objects attract one another. While the gravitational force is the force of attraction which keeps two bodies in the universe bonded together.

Factors Affecting the Gravitational Force Between Two Bodies:

• The gravitational force of attraction between two bodies is directly proportional to the product of masses. If the distance between two masses is constant, then an increase in the mass of one of the two or of both increases the gravitational force of attraction between the two bodies.
• The gravitational force of attraction between two bodies is inversely proportional to the square distance between the two bodies. If the masses are kept constant, then the increase in distance between the two bodies decreases the gravitational force of attraction between the two bodies.
• The gravitational force between two masses is independent of the medium between the two masses.
• The gravitational force between two bodies does not depend upon the presence or the absence of other bodies.

S.I. Unit of G:

By Newton’s law of gravitation

The SI unit of constant of gravitation is N m² kg^-2 and c.g.s. unit is dyne cm² g^-2.

Dimensions of G:

By Newton’s law of gravitation

Hence the dimensions of universal gravitation constant are [M^-1 L³ T^-2]

Definition of G:

The gravitational force between two point masses ‘m₁’ and ‘m₂’ separated by distance ‘r; is given by

Let r = 1 unit, m₁ = m₂ = 1 unit, then G = F

Hence, the universal gravitational constant is the numerical value of the force between two unit masses kept at a unit distance from each other.

The value of G is very small and gravitational forces are small unless the masses of the two attracting bodies are large. If the value of G becomes 100 times its present value, then the earth’s attraction would be so large that we would be crushed to the earth. If the value of G becomes 1/100 times its present value, then we would be able to jump from a multi-story building.

Principle of Superposition of Forces:

Next Topic: Numerical Problems on Newton’s Law of Gravitation

Science > Physics > Gravitation >Newton’s Law of Gravitation

The post Newton’s Law of Gravitation appeared first on The Fact Factor.