Gravitational Constant — Revision Notes

⚡ 30-Second Revision

Definition: — Universal Gravitational Constant, $G$.
Formula: — $F = G \frac{m_1 m_2}{r^2}$
Value: — $G approx 6.674 \times 10^{-11} N cdot m^2/kg^2$
Units: — $N cdot m^2/kg^2$ or $m^3/(kg cdot s^2)$
Dimensions: — $[M^{-1} L^3 T^{-2}]$
Nature: — Universal, scalar, independent of mass, distance, medium.
Discovery: — First measured by Henry Cavendish (1798) using a torsion balance.
Key Distinction: — Not to be confused with $g$ (acceleration due to gravity), which is variable.

2-Minute Revision

The Universal Gravitational Constant, $G$, is a fundamental constant in physics, central to Newton's Law of Universal Gravitation ($F = G \frac{m_1 m_2}{r^2}$). Its approximate value is $6.674 \times 10^{-11} N cdot m^2/kg^2$.

Key aspects for NEET include its units ($N cdot m^2/kg^2$ or $m^3/(kg cdot s^2)$) and dimensional formula ($[M^{-1} L^3 T^{-2}]$), which are frequently tested. $G$ is a universal constant, meaning its value is invariant across the cosmos and does not depend on the masses involved, the distance between them, or the medium.

It is a scalar quantity. Crucially, distinguish $G$ from $g$ (acceleration due to gravity); $G$ is constant, while $g$ varies with location and celestial body. Henry Cavendish first measured $G$ experimentally using a torsion balance.

Its small value explains why gravitational forces are only significant for massive astronomical objects.

5-Minute Revision

The Universal Gravitational Constant, $G$, is the proportionality constant in Newton's Law of Universal Gravitation, $F = G \frac{m_1 m_2}{r^2}$. This law describes the attractive force between any two objects with mass.

$G$ quantifies the inherent strength of this gravitational interaction. Its accepted value is approximately $6.674 \times 10^{-11} N cdot m^2/kg^2$. This extremely small value implies that gravity is a weak force, only becoming significant for objects with enormous masses, like planets and stars.

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Universality: — $G$ is a true constant, independent of the masses, their separation, the medium between them, or any environmental factors like temperature or pressure. This is a common conceptual trap.
2
Scalar Nature: — $G$ is a scalar quantity; it only has magnitude. The gravitational force, however, is a vector.
3
Units: — Derived from $G = \frac{F r^2}{m_1 m_2}$, its SI units are $N cdot m^2/kg^2$. In fundamental SI units, this translates to $m^3/(kg cdot s^2)$.
4
Dimensional Formula: — From its units, the dimensional formula is $[M^{-1} L^3 T^{-2}]$. This is a high-yield topic for MCQs.
5
Experimental Determination: — Henry Cavendish first measured $G$ in 1798 using a torsion balance, an experiment famously dubbed 'weighing the Earth'.

Crucial Distinction (G vs. g): Do not confuse $G$ with $g$ (acceleration due to gravity). $G$ is a universal constant, while $g$ is variable, depending on the mass and radius of the celestial body and the altitude. For example, $g$ on Earth is $approx 9.8 m/s^2$, but $G$ is always $6.674 \times 10^{-11} N cdot m^2/kg^2$. The relationship is $g = G \frac{M}{R^2}$.

Example: If two masses of $5,\text{kg}$ each are $0.5,\text{m}$ apart, the force is: $F = (6.67 \times 10^{-11}) \frac{(5)(5)}{(0.5)^2} = (6.67 \times 10^{-11}) \frac{25}{0.25} = (6.67 \times 10^{-11}) \times 100 = 6.67 \times 10^{-9},\text{N}$. This small value reinforces the weakness of gravity between everyday objects.

Prelims Revision Notes

The Universal Gravitational Constant, $G$, is a fundamental constant in physics, appearing in Newton's Law of Universal Gravitation: $F = G \frac{m_1 m_2}{r^2}$.

Key Facts to Remember:

Value: — $G approx 6.674 \times 10^{-11} N cdot m^2/kg^2$. (Approximate value $6.67 \times 10^{-11}$ is often sufficient).
Units: — Derived from $G = \frac{F r^2}{m_1 m_2}$.

* SI Units: $N cdot m^2/kg^2$. * In fundamental SI units: $m^3/(kg cdot s^2)$.

Dimensional Formula: — $[G] = [M^{-1} L^3 T^{-2}]$. This is a very high-yield topic.

* Derivation: $[F] = [M L T^{-2}]$, $[r^2] = [L^2]$, $[m^2] = [M^2]$. So, $[G] = \frac{[M L T^{-2}] [L^2]}{[M^2]} = [M^{-1} L^3 T^{-2}]$.

Nature:

* Universal: Its value is constant throughout the universe. It does NOT depend on: * Masses of the objects. * Distance between them. * Medium between the objects (e.g., vacuum, air, water). * Temperature, pressure, or any other physical conditions. * Scalar Quantity: It has magnitude only, no direction.

Discovery/Measurement: — First accurately measured by Henry Cavendish in 1798 using a torsion balance experiment.
Distinction from 'g' (acceleration due to gravity):

* $G$: Universal constant, fixed value, scalar. * $g$: Variable, depends on location/planet, vector (acceleration). * Relationship: $g = G \frac{M}{R^2}$ (where $M$ is planet mass, $R$ is planet radius).

Significance of Small Value: — Its tiny magnitude means gravitational forces are only significant for very large masses (planets, stars). For everyday objects, the force is negligible.

Common Traps:

Confusing $G$'s independence from medium/mass/distance with $g$'s variability.
Incorrectly recalling units or dimensions.
Attributing $G$'s measurement to Newton.

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Great Men Love To Derive Units: Gravitational constant, Mass (inverse), Length (cubed), Time (inverse squared), Dimensions, Units.