Physics and Mathematics
Inertial Mass and Gravitational Mass
1. Concept Overview
Every object possesses mass, but mass can be defined in two different ways depending on how it behaves under force and gravity:
- Inertial Mass — Determines how much a body resists acceleration when a force is applied.
- Gravitational Mass — Determines how strongly a body interacts with the gravitational field.
Although conceptually different, experiments show that these two masses are equivalent — a principle that forms the basis of Einstein’s General Theory of Relativity.
2. Explanation and Mathematical Derivation
(a) Inertial Mass ([m_i])
According to Newton’s Second Law of Motion,
[
F = m_i a
]
Thus,
[
m_i = \dfrac{F}{a}
]
The inertial mass is a measure of the inertia (resistance to change in motion) of a body.
A larger inertial mass means more resistance to acceleration.
(b) Gravitational Mass ([m_g])
A body of mass [m_g] in a gravitational field of another mass [M] experiences a gravitational force given by Newton’s Law of Gravitation:
[
F = G \dfrac{M m_g}{r^2}
]
The gravitational mass determines the strength of the gravitational attraction between two bodies.
(c) Equivalence of [m_i] and [m_g]
For a body falling freely near Earth’s surface:
[\text{Gravitational Force}] [= \text{Inertial Force}]
[
G \dfrac{M_E m_g}{R_E^2} = m_i g
]
[\Rightarrow \dfrac{m_g}{m_i}] [= \dfrac{g R_E^2}{G M_E} = 1]
Hence,
[
m_g = m_i
]
Thus, inertial mass and gravitational mass are equivalent — verified experimentally by Eötvös and Galileo.
3. Dimensions and Units
| Type of Mass | Symbol | Dimensions | SI Unit |
|---|---|---|---|
| Inertial Mass | [m_i] | [M^1] | kilogram (kg) |
| Gravitational Mass | [m_g] | [M^1] | kilogram (kg) |
4. Key Features
- Both [m_i] and [m_g] represent the same physical quantity — mass.
- [m_i] defines how much force is required to accelerate a body.
- [m_g] defines how strongly the body interacts gravitationally.
- Experiments show their ratio [\dfrac{m_g}{m_i}] = 1.
- The equivalence principle is the cornerstone of General Relativity.
- In a uniform gravitational field, all bodies fall with the same acceleration.
- [m_i] is measured using non-gravitational forces, while [m_g] is measured using gravitational attraction.
- Both are scalar quantities.
5. Important Formulas to Remember
| Quantity | Symbol | Formula | Remarks |
|---|---|---|---|
| Inertial Mass | [m_i] | [m_i = \dfrac{F}{a}] | From [F = m a] |
| Gravitational Mass | [m_g] | [F = G \dfrac{M m_g}{r^2}] | From law of gravitation |
| Weight | [W] | [W = m_g g] | On Earth’s surface |
| Relation | — | [m_i = m_g] | Equivalence principle |
| Acceleration | [a] | [a = \dfrac{G M_E}{R_E^2}] | For freely falling body |
6. Conceptual Questions with Solutions
1. What is inertial mass?
It is the measure of the resistance of a body to change in its motion when a force is applied: [m_i = \dfrac{F}{a}].
2. What is gravitational mass?
It determines the strength of the gravitational interaction between two bodies: [F = G \dfrac{M m_g}{r^2}].
3. How are inertial and gravitational masses related?
Experiments show that [m_i = m_g].
4. Why does a feather and a stone fall together in vacuum?
Because [m_g/m_i = 1] for both, hence they experience the same acceleration [g].
5. What is the equivalence principle?
It states that gravitational and inertial masses are identical, forming the basis of Einstein’s General Relativity.
6. How can we experimentally verify the equivalence?
By Eötvös’ torsion balance experiment — showing identical acceleration for different substances.
7. Which mass is used in Newton’s second law?
Inertial mass ([m_i]).
8. Which mass appears in the law of gravitation?
Gravitational mass ([m_g]).
9. Why are both masses equal numerically?
Because all objects fall with the same acceleration under gravity.
10. Does gravitational mass depend on the nature of the substance?
No, it depends only on the total amount of matter.
11. How do astronauts test equivalence in space?
By comparing accelerations of different materials in free fall.
12. Is gravitational mass the same in all gravitational fields?
Yes, it’s an intrinsic property of the body.
13. What happens if [m_g ≠ m_i]?
Then different bodies would fall with different accelerations — contradicting observations.
14. Why does a heavier object not fall faster than a lighter one?
Because gravitational and inertial masses are proportional.
15. How is this principle used in modern physics?
It underpins Einstein’s idea that gravity and acceleration are indistinguishable locally.
7. FAQ / Common Misconceptions
1. Inertial mass and gravitational mass are different quantities.
❌ They are conceptually different but numerically equal.
2. Heavier objects fall faster than lighter ones.
❌ False. All bodies fall with the same acceleration in vacuum.
3. Inertial mass depends on gravity.
❌ It’s independent of the gravitational field.
4. Gravitational mass is a vector.
❌ Both are scalars.
5. The equivalence of [m_i] and [m_g] is an assumption.
❌ It is experimentally verified.
6. Only gravitational mass affects weight.
✅ True, since [W = m_g g].
7. Inertial mass changes with velocity.
Relativistically, yes — at very high speeds, effective inertia increases.
8. The value of [m_g/m_i] differs for gases and solids.
❌ False. It’s always 1.
9. Einstein rejected the equivalence principle.
❌ He extended it to build General Relativity.
10. Weightlessness occurs because [m_g] becomes zero.
❌ Weightlessness occurs because there’s no normal reaction, not because [m_g = 0].
8. Practice Questions (With Step-by-Step Solutions)
Q1. A force of 20 N produces an acceleration of 5 m/s² in a body. Find its inertial mass.
[m_i = \dfrac{F}{a}] [= \dfrac{20}{5} = 4kg]
Q2. If the gravitational force between Earth and a 2 kg mass is [19.6N], find [m_g].
[F = m_g g \Rightarrow m_g] [= \dfrac{F}{g}] [= \dfrac{19.6}{9.8} = 2kg]
Q3. A 1 kg stone and a 10 kg iron ball fall freely in vacuum. Compare their accelerations.
[
a = g = 9.8,m/s^2
]
Answer: Both fall with equal acceleration.
Q4. Show that [m_i = m_g] using Earth’s gravitational field parameters.
[G \dfrac{M_E m_g}{R_E^2}] [= m_i g \Rightarrow \dfrac{m_g}{m_i}] [= \dfrac{g R_E^2}{G M_E}] [= 1]
Hence, [m_g = m_i].
Q5. What would happen if [m_g \neq m_i]?
Then different bodies would fall with different accelerations — violating Galileo’s observation and Newton’s laws.
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