Physics and Mathematics
Characteristics of Gravitational Forces
1. Concept Overview
The gravitational force is the force of mutual attraction between any two objects that possess mass. It is one of the fundamental forces of nature and governs the motion of celestial bodies, satellite orbits, tides, and much more.
Every mass in the universe exerts an attractive force on every other mass, however small or large. Despite being the weakest among all fundamental forces, its range is infinite, and it acts on all masses without exception.
2. Explanation and Mathematical Derivation
From Newton’s Law of Gravitation,
[
F = G \dfrac{m_1 m_2}{r^2}
]
We can identify several characteristics of this force based on its mathematical form:
(i) Dependence on Masses
The gravitational force is directly proportional to the product of the interacting masses:
[
F \propto m_1 m_2
]
If either mass increases, the gravitational attraction also increases.
(ii) Dependence on Distance
It is inversely proportional to the square of the distance between their centers:
[
F \propto \dfrac{1}{r^2}
]
If distance doubles, force becomes one-fourth.
(iii) Direction of Force
It acts along the line joining the centers of the two bodies. Hence, it is a central force.
(iv) Attractive Nature
The force is always attractive, never repulsive, because mass is always positive.
(v) Action and Reaction Pair
According to Newton’s Third Law, the forces between two bodies are equal in magnitude and opposite in direction:
[
\vec{F}_{12} = -\vec{F}_{21}
]
(vi) Long-Range Force
Gravitational force decreases with distance but never becomes zero. It acts even over astronomical distances.
(vii) Independent of Medium
Unlike electric or magnetic forces, gravitational force is not affected by the presence of any medium. It acts equally well in vacuum.
(viii) Universal Constant ( G )
The proportionality constant ( G ) is the same throughout the universe.
[
G = 6.674 \times 10^{-11} \text{N·m}^2/\text{kg}^2
]
(ix) Conservative Nature
Gravitational force is a conservative force, meaning the work done depends only on initial and final positions, not on the path.
(Refer to Work and Energy for a detailed discussion on conservative forces.)
(x) Central and Inverse-Square Law
Being both central and inverse-square, it gives rise to elliptical orbits and stable planetary motion (Refer to Differentiation for derivations involving inverse-square forces).
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Gravitational Force | [ F ] | N (newton) | ([M^1 L^1 T^{-2}]) |
| Gravitational Constant | [ G ] | N·m²/kg² | ([M^{-1} L^3 T^{-2}]) |
| Distance | [ r ] | m | ([L]) |
| Mass | [ m ] | kg | ([M]) |
4. Key Features
- Acts between any two masses, irrespective of their state (solid, liquid, or gas).
- Always attractive in nature.
- Mutual and obeys Newton’s third law.
- Central and conservative in behavior.
- Infinite range (though weak at large distances).
- Independent of medium.
- Follows the inverse-square law.
- Provides the binding force for planetary systems, stars, and galaxies.
- Does not depend on the shape or size of the bodies (only on mass and distance).
- Its effect is superimposable — the net gravitational force on a body is the vector sum of all forces acting on it (Principle of Superposition).
5. Important Formulas to Remember
| Concept | Formula | Description |
|---|---|---|
| Gravitational Force | [ F = G \dfrac{m_1 m_2}{r^2} ] | Attraction between two point masses |
| Vector Form | [ \vec{F}_{12} = -G \dfrac{m_1 m_2}{r^2} \hat{r}_{12} ] | Directional and attractive |
| Force Ratio | [ \dfrac{F_1}{F_2} = \dfrac{r_2^2}{r_1^2} ] | Change with distance |
| Dimensional Formula of ( G ) | ( [M^{-1}L^3T^{-2}] ) | Dimensional representation |
| Work Done by Gravitational Force | [ W = G \dfrac{m_1 m_2}{r_1} – G \dfrac{m_1 m_2}{r_2} ] | Proves conservative nature |
6. Conceptual Questions with Solutions
1. Why is gravitational force called a central force?
Because it always acts along the line joining the centers of two interacting bodies.
2. Is gravitational force always attractive?
Yes, since mass is always positive, gravitational force can never be repulsive.
3. Does gravitational force depend on the medium between the bodies?
No, gravitational force is independent of any medium and acts in vacuum as well.
4. What is meant by a long-range force?
A long-range force acts even at very large distances, like between Earth and the Sun.
5. What is meant by a conservative force?
A conservative force does work independent of path; only initial and final positions matter.
6. Does the gravitational force follow the inverse-square law?
Yes. If the distance doubles, force becomes one-fourth; if halved, force becomes four times.
7. What ensures the stability of planetary orbits?
The central and inverse-square nature of the gravitational force ensures stable elliptical orbits.
8. Is gravitational force stronger than electromagnetic force?
No, gravitational force is much weaker (by about \(10^{36}\) times) than the electromagnetic force.
9. What determines the direction of gravitational force?
The line joining the centers of the two masses determines the force’s direction.
10. Why is gravitational force said to obey the superposition principle?
Because the net gravitational force on a body is the **vector sum** of all individual gravitational forces acting on it.
11. Is gravitational force mutual?
Yes, both bodies exert equal and opposite forces on each other.
12. Does gravitational force depend on the state of motion of the bodies?
No, it depends only on masses and their distance, not their motion.
13. Why can we ignore gravitational force between small objects?
Because the force is extremely small compared to other forces like friction or electrostatic force.
14. What happens to gravitational force if one of the masses becomes zero?
Force becomes zero since \( F \propto m_1 m_2 \).
15. What proves that gravitational force is conservative?
The work done in moving a mass in a closed path under gravity is zero.
7. FAQ / Common Misconceptions
1. Gravitational force acts only on Earth — True or False?
**False.** It acts everywhere between any two masses in the universe.
2. The gravitational force can be repulsive.
**False.** It is always attractive.
3. The gravitational constant \( G \) changes from place to place.
**False.** \( G \) is universal and constant everywhere.
4. Gravitational force depends on shape of the body.
**False.** It depends only on total mass and distance.
5. Gravitational force requires contact between bodies.
**False.** It is a non-contact force acting through space.
6. The gravitational force can be completely shielded.
**False.** There’s no known material that blocks gravitational attraction.
7. Gravitational force becomes zero in space.
**False.** It only becomes weaker with distance but never zero.
8. Heavy bodies fall faster due to greater gravitational force.
**False.** All bodies fall with the same acceleration (neglecting air resistance).
9. The gravitational force is always the same on all objects.
**False.** It depends on the masses and distance between them.
10. The gravitational force is stronger on heavier planets because of their mass alone.
**Partly true.** Heavier planets have more mass, but the surface gravity also depends on the square of their radius.
8. Practice Questions (Step-by-Step Solutions)
Q1. State any two characteristics of gravitational force.
Solution:
- It is always attractive.
- It acts along the line joining the centers of the two masses.
Q2. The gravitational force between two bodies is [ 10 \text{N} ] at a distance of [ 2 \text{m} ]. What will be the force if distance becomes [ 4 \text{m}]
Solution:
[F’ = F \left( \dfrac{r_1}{r_2} \right)^2] [= 10 \left( \dfrac{2}{4} \right)^2] [= 10 \times \dfrac{1}{4} = 2.5 \text{N}]
Q3. What type of force is gravitational force and why?
Solution:
It is a central and conservative force because it acts along the line joining centers and the work done depends only on initial and final positions.
Q4. If the distance between two masses is reduced to half, how does gravitational force change?
Solution:
[F’ = F \left( \dfrac{r_1}{r_2} \right)^2] [= F \left( \dfrac{1}{0.5} \right)^2 = 4F]
Force becomes four times the original.
Q5. What ensures that gravitational force obeys Newton’s Third Law?
Solution:
The force on [ m_1 ] due to [ m_2 ] is equal and opposite to the force on [ m_2 ] due to [ m_1 ]:
[\vec{F}_{12} = -\vec{F}_{21}]
Hence, gravitational forces are universal, attractive, conservative, and central — forming the basis of celestial mechanics and everyday gravity.
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