Universal Law of Gravitation — Revision Notes

⚡ 30-Second Revision

Universal Law of Gravitation: — $F = G \frac{m_1 m_2}{r^2}$
Universal Gravitational Constant (G): — $6.674 \times 10^{-11} ,\text{N m}^2/\text{kg}^2$ (scalar, universal, independent of medium)
Acceleration due to Gravity (g): — $g = G \frac{M}{R^2}$ (vector, varies with location, depends on planet's mass $M$ and radius $R$)
Nature of Force: — Always attractive, acts along the line joining centers.
Inverse Square Law: — $F propto 1/r^2$. If $r$ doubles, $F$ becomes $F/4$.
Superposition Principle: — Net force is vector sum of individual forces.
Weakest Fundamental Force.

2-Minute Revision

The Universal Law of Gravitation describes the attractive force between any two objects with mass. This force is directly proportional to the product of their masses ($m_1m_2$) and inversely proportional to the square of the distance ($r^2$) between their centers, given by $F = G \frac{m_1 m_2}{r^2}$.

The constant $G$ is the Universal Gravitational Constant, a tiny value ($6.674 \times 10^{-11} ,\text{N m}^2/\text{kg}^2$) that makes gravity the weakest of the four fundamental forces. It is a scalar, universal, and independent of the medium.

Gravitational force is always attractive and acts along the line connecting the centers of the masses. For multiple masses, the net force on any one mass is the vector sum of individual forces (superposition principle).

A key application is understanding acceleration due to gravity ($g = G \frac{M}{R^2}$), which varies with the celestial body's mass ($M$) and radius ($R$), and also with altitude, unlike the constant $G$.

Remember to correctly apply the inverse square law for distance changes and use vector addition for multiple forces.

5-Minute Revision

Newton's Universal Law of Gravitation is fundamental: $F = G \frac{m_1 m_2}{r^2}$. This formula tells us that every mass attracts every other mass. The force ($F$) is directly proportional to the product of the masses ($m_1, m_2$) and inversely proportional to the square of the distance ($r$) between their centers.

The constant $G$, the Universal Gravitational Constant, is a fixed value ($6.674 \times 10^{-11} ,\text{N m}^2/\text{kg}^2$) that quantifies gravity's inherent weakness. It's crucial to remember that $G$ is universal, scalar, and independent of the medium.

Gravitational force is always attractive, pulling objects together. It's a central force, acting along the line joining the centers of the masses. When multiple masses are present, the total force on any one mass is found by vectorially adding the individual forces from all other masses (superposition principle).

One of the most important consequences is the acceleration due to gravity ($g$). On a planet of mass $M$ and radius $R$, $g = G \frac{M}{R^2}$. Unlike $G$, $g$ is not constant; it varies with altitude (decreasing as $1/(R+h)^2$) and depends on the specific celestial body. For example, if Earth's radius were halved with constant mass, $g$ would become $4g$.

Worked Example: Two masses, $2,\text{kg}$ and $3,\text{kg}$, are separated by $0.5,\text{m}$. Calculate the gravitational force between them. Given: $m_1 = 2,\text{kg}$, $m_2 = 3,\text{kg}$, $r = 0.5,\text{m}$, $G = 6.67 \times 10^{-11} ,\text{N m}^2/\text{kg}^2$. $F = G \frac{m_1 m_2}{r^2} = (6.67 \times 10^{-11}) \frac{(2)(3)}{(0.5)^2} = (6.67 \times 10^{-11}) \frac{6}{0.25} = (6.67 \times 10^{-11}) \times 24 = 160.08 \times 10^{-11},\text{N} = 1.60 \times 10^{-9},\text{N}$.

For NEET, focus on ratio problems, vector addition for symmetric configurations, and the distinction between $G$ and $g$. Always consider the distance from the center of the body for calculations involving planets.

Prelims Revision Notes

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Newton's Law of Gravitation: — $F = G \frac{m_1 m_2}{r^2}$. This is the magnitude of the attractive force between two point masses $m_1$ and $m_2$ separated by distance $r$.
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Universal Gravitational Constant (G):

* Value: $6.674 \times 10^{-11} ,\text{N m}^2/\text{kg}^2$. * Nature: Scalar quantity. * Universality: Its value is constant throughout the universe. * Medium: Independent of the medium between the masses. * Weakness: It's the weakest of the four fundamental forces.

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Properties of Gravitational Force:

* Always attractive. * Acts along the line joining the centers of the two masses (central force). * Obeys Newton's Third Law (action-reaction pair). * Independent of the presence of other bodies (superposition principle).

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Acceleration due to Gravity (g):

* Definition: Acceleration experienced by an object due to a celestial body's gravity. * Formula: $g = G \frac{M}{R^2}$, where $M$ is the mass of the celestial body and $R$ is its radius. * Variability: $g$ is NOT constant.

It varies with: * Altitude: Decreases with height $h$ above surface: g_h = g left(\frac{R}{R+h}\right)^2. If $h ll R$, g_h approx g left(1 - \frac{2h}{R}\right). * Depth: Decreases with depth $d$ below surface: g_d = g left(1 - \frac{d}{R}\right).

At center ($d=R$), $g_d = 0$. * Shape of Earth: Max at poles, min at equator due to Earth's bulge and rotation. * Mass and Radius of Planet: Directly proportional to planet's mass, inversely proportional to square of its radius.

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Inverse Square Law: — The force is proportional to $1/r^2$. If distance doubles, force becomes $1/4$. If distance halves, force becomes $4$ times.
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Superposition Principle: — For a system of multiple particles, the net gravitational force on any particle is the vector sum of the forces exerted by all other particles. $vec{F}_{net} = sum vec{F}_i$.
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Comparison with Electrostatic Force: — Both are inverse square laws, but gravity is always attractive, acts on mass, is much weaker, and is independent of the medium. Electrostatic force can be attractive or repulsive, acts on charge, is much stronger, and depends on the medium.

Vyyuha Quick Recall

To remember the formula $F = G \frac{m_1 m_2}{r^2}$: For Gravity, Many Masses Radiate Strongly. (F = Force, G = Gravitational constant, M = Mass, R = Radius/distance, S = Square - for $r^2$)