Gravitation: Gravitational Force , Field, and Potential Energy

Gravitation, also known as gravitational attraction, is the attraction of all masses in the universe to one another.

• Sir Isaac Newton proposed that all particles in the universe attract each other.
• The force of attraction between any two bodies of this universe is called Gravitation or Gravitational force.
• Gravity is the force that pulls objects toward the center of a planet or other body.
• The force of gravity keeps all of the planets in orbit around the sun.
• Gravity exists in everything that has mass.
• Gravity is stronger for heavier objects.
• Gravity gets weaker with distance as well.
• The stronger the gravitational force between two objects, the closer they are to each other.

Key Terms: Gravitational force, Gravitation, Gravity, Acceleration due gravity, Gravitational field, Gravitational potential energy, Weightlessness, Orbital velocity, Kepler’s law, Law of periods.

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Gravitational Force

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Gravitational force, also known as the force of gravity is the force with which the objects with mass attract towards each other.

• Any object having mass exerts a gravitational force on all other mass objects.
• For example, there is a gravitational pull between you and all objects in your surroundings.
• When the masses of two objects are larger, the gravitational force between them is larger.
• The gravitational force between two objects is also affected by the separation between their centers.
• The greater the distance between objects, the weaker the force.

Characteristics of Gravitational Force

The following are the characteristics of gravitational force

• The gravitational force between two bodies forms an action and reaction pair.
• Gravitational force is a central force i.e. it always acts along the line joining the centers of the two bodies.
• It is independent of the nature of the intervening medium.
• It is a long-range force.
• It is a conservative force.

Gravitation Video Explanation

Gravitation Detailed Video Explanation

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History of Gravitational Theory

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The history of the theory of gravitation is given below

• According to Ptolemy “The Sun, the moon, and planets were in uniform motion in circles called epicycles with the fixed earth at the center”.
• This is known as Geo-centric Theory.
• According to Aryabhatta, the earth moves around the sun along with the other planets.
• Copernicus proposed the sun-centered theory according to which the Earth and all other planets move around the sun.
• This theory is known as the Helio-centric theory.
• After the acceptance of Copernicus's theory, Tycho Brahe made an accurate measurement of the position of stars and planets.
• His data was analyzed by Johannes Kepler who concluded that orbits of the planets are not circular, but elliptical.
• He proposed three laws known as Kepler’s laws.

Gravitation

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Newton’s Law of Gravitation

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Newton’s law of gravitation is also known as the Universal law of gravitation.

According to this law,

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

Mathematically

\(F=G\frac {m_1m_2}{r^2}\)

Where

• F is the gravitational force
• m₁ and m₂ are the masses of the two bodies
• G is known as the Gravitational constant
• r is the distance between the two bodies.

This law holds good irrespective of the nature of two objects at all places and at all times throughout the universe.

• That is why it is known as the Universal law of gravitation.
• As F varies inversely as a square of distance, it is also known as the inverse square law force.

Gravitational Constant

The Gravitational constant is defined as the force of attraction between two bodies of unit masses separated by unit distance.

• It is denoted by G.
• The value of the gravitation constant is G = 6.67 x 10^-11 N m² kg^-2
• The dimension formula of G is [M^-1L³T^-2].

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Newton’s Law of Gravitation in Vector Form

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Consider two bodies of masses m₁ and m₂ separated by distance r.

The force acting on m₂ due to m₁ is given by The force acting on m₂ due to m₁ is given by Where \(\hat r_{21}\) and \(\hat r_{12}\) are the unit vectors. But \(\hat r_{12} = -\hat r_{21}\) Therefore

The Gravitational force between two bodies is equal in magnitude but opposite in direction. Hence Gravitational force is Newtonian in nature.

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Principle of Superposition of Gravitational Forces

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According to the principle of superposition of gravitational forces

The resultant gravitational force on a point mass due to the number of point masses is the vector sum of the gravitational forces exerted on that point mass due to a number of point masses.

Mathematically

\(\vec F_{net}=\vec F_{12}+\vec F_{13}+\vec F_{14}+......+\vec F_{1n}\)

Where

• F_net is the net force acting on particle 1.
• F₁₂ is the force acting on particle 1 due to particle 2.
• Similarly, F₁₃, F₁₄, ……, F_1n are the forces acting on particle 1 due to particles 2, 3,….n.

Newton's law of gravitation only applies to interactions between two particles; if the system has 'n' particles, the number of such interactions is

n(n - 1)/2

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Acceleration Due to Gravity

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The acceleration produced in the motion of the body under the effect of gravity is known as acceleration due to gravity.

• It is denoted by g.
• It is a vector quantity.
• The SI unit of acceleration due to gravity is m/s².
• The value of g on the surface of the earth is 9.8 m/s².

The formula of acceleration due to gravity on the surface of a planet is given by

\(g=\frac{GM_P}{R_P^2}\)

Where

• G is the Gravitational constant
• M_P is the mass of the planet
• R_P is the radius of the planet.

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Gravitational Potential Energy

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When a body of mass (m) is displaced from infinity to a point under the gravitational influence of a source mass (M) without being accelerated, the amount of work done in displacing it into the source field is stored as potential energy.

• This is referred to as gravitational potential energy.
• Gravitational potential energy is the energy associated with the position of the particle.
• In the work, energy, and power chapter, U = mgh, is valid only when the particle is near the surface of the earth.
• Gravitational potential energy (U) is a scalar quantity and is measured in joules.

The formula of Gravitational potential energy for a system of two masses separated by distance r is given by

\(U=-G\frac {m_1m_2}{r^2}\)

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Derivation of Newton’s Law of Gravitation From Kepler’s Law

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Assume a test mass rotates around a source mass in a nearly circular orbit with radius 'r' and constant angular speed (ω). The centripetal force acting on the test mass causes circular motion is given by

F = mrω²

But ω = 2π/T

Where T is the period of the revolution.

⇒ F = mr (2π/T)²

⇒ F = mr (4π²/T²)    ….(i)

According to Kepler’s Third law, we have

T² ∝ r³

⇒ T² = kr³

Where k = 4π²/GM

Using the above equation in equation (i), we get

F = mr (4π²/kr³)

⇒ F = GMm/r²

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Things to Remember

• Gravitation is the attraction of all masses in the universe to one another.
• It is also known as gravitational attraction.
• The force of gravity is the force with which the objects with mass attract towards each other.
• According to Newton’s law of gravitation, F = G (m₁m₂/r²).
• Gravitational potential energy is the energy associated with the position of the particle.
• The formula of Gravitational potential energy is U = – G (m₁m₂/r).
• The acceleration produced in the motion of the body under the effect of gravity is known as acceleration due to gravity.
• The value of g on the surface of the earth is 9.8 m/s².
• The value of the gravitation constant is G = 6.67 x 10^-11 N m² kg^-2