Notes : Newton's Law of Universal Gravitation and Gravitational Constant (G) S.I Unit , Dimension

Newton's Law of Universal Gravitation and Universal Gravitational Constant (G)

Newton's Law of Universal Gravitation

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

Mathematical Expression:

$F \propto m_{1}m_{2}$

$F  \propto \frac{1}{r^{2}}$

$F = G \frac{m_{1}m_{2}}{r^{2}}$

Where:

• F = Gravitational force between the two bodies
• m₁, m₂ = Masses of the two bodies
• r = Distance between their centers
• G = Universal Gravitational Constant

Characteristics of Gravitational Force

• It acts between every pair of masses in the universe.
• It is always attractive in nature.
• It acts along the line joining the centers of the two bodies.
• It obeys the inverse square law.
• Its magnitude depends on masses and distance between the bodies.

Universal Gravitational Constant (G)

Definition

The Universal Gravitational Constant (G) is defined as the gravitational force of attraction between two bodies of mass 1 kg each placed 1 m apart in vacuum.

We know , 

$F= G \frac{m_{1}m_{2}}{r^{2}}$

If $m_1 = m_2 = 1 kg , r= 1m$

Then G = F

Value of G

G = 6.674 × 10^-11 N·m²/kg²

Importance of G

• G is a universal constant.
• Its value remains the same everywhere in the universe.
• It helps calculate the gravitational force between any two bodies.

Derivation of SI Unit of G

From Newton's law:

$F = \frac{G m_{1} m_{2}}{r^{2}}$

$G = \frac{F r^{2}}{m_{1} m_{2}}$

Substituting SI units:

• Force = N
• Distance = m
• Mass = kg

[G] = (N × m²)/kg²

SI Unit of G

N·m²/kg²

Derivation of Dimensional Formula of G

$G = \frac{F r^{2}}{m_{1} m_{2}}$

Dimensions of force:

$[F] = [MLT^{-2}]$

$[G] =\frac{ [MLT^{-2}] × [L^{2}] }{[M^{2}]}$

$[G] = [MLT^{-2}] × [L^{2}] × [M^{2}]$

$[G] = [M^{-1}L^{3}T^{-2}]$

Dimensional Formula of G

$[G] = [M^{-1}L^{3}T^{-2}]$

Summary

Quantity	Expression
Newton's Law of Gravitation	F = Gm₁m₂/r²
Value of G	6.674 × 10-11 N·m²/kg²
SI Unit of G	N·m²/kg²
Dimensional Formula of G	[M-1L3T-2]
Nature of Gravitational Force	Always Attractive

Frequently Asked Questions (FAQs)

Q1. State Newton's Law of Universal Gravitation.

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

Q2. What is the value of Universal Gravitational Constant (G)?

G = 6.674 × 10^-11 N·m²/kg²

Q3. What is the SI unit of G?

N·m²/kg²

Q4. What is the dimensional formula of G?

[M^-1L³T^-2]

Q5. Why is G called a universal constant?

Because its value remains the same everywhere in the universe.

Quiz Questions (MCQs)

1. Newton's law of gravitation states that gravitational force is directly proportional to:

1. Distance between the bodies
2. Square of distance between the bodies
3. Product of the masses of the bodies
4. Sum of the masses of the bodies

Answer: C) Product of the masses of the bodies

2. Gravitational force between two bodies is inversely proportional to:

1. Distance
2. Square of distance
3. Cube of distance
4. Product of masses

Answer: B) Square of distance

3. The SI unit of Universal Gravitational Constant (G) is:

1. N/kg
2. N·m/kg²
3. N·m²/kg²
4. kg·m²/N

Answer: C) N·m²/kg²

4. The dimensional formula of G is:

1. [MLT⁻²]
2. [M⁻¹L³T⁻²]
3. [M⁰L²T⁻²]
4. [ML²T⁻¹]

Answer: B) [M⁻¹L³T⁻²]

5. Gravitational force is always:

1. Repulsive
2. Attractive
3. Attractive and Repulsive
4. Zero

Answer: B) Attractive

True / False Questions

• Gravitational force is always attractive. (True)
• G depends on the nature of the medium. (False)
• The value of G is the same everywhere in the universe. (True)
• Gravitational force follows the inverse square law. (True)
• The SI unit of G is kg·m/s². (False)

Short Answer Questions

1. State Newton's Law of Universal Gravitation.
2. Define Universal Gravitational Constant (G).
3. Write the SI unit of G.
4. Write the dimensional formula of G.
5. Why is G called a universal constant?
6. Who experimentally measured the value of G?
7. What is the value of G?
8. Why is gravitational force called a central force?

Numerical Questions

1. Two masses of 2 kg and 5 kg are placed 1 m apart. Calculate the gravitational force between them.
2. Calculate the gravitational force between two masses of 10 kg each separated by a distance of 2 m.
3. If the distance between two masses is tripled, by what factor does the gravitational force change?
4. If both masses are doubled while the distance remains unchanged, what happens to the gravitational force?
5. A force of 6.67 × 10^-11 N acts between two bodies of mass 1 kg each placed 1 m apart. Identify the physical constant represented.

HOTS Questions

1. Why don't we feel the gravitational attraction between two persons standing close to each other?
2. Can gravitational force ever become repulsive? Explain.
3. Why is the value of G extremely small?
4. Why does the Moon not fall directly onto the Earth despite being attracted by Earth's gravity?
5. How would life be affected if the value of G suddenly became twice its present value?