# 15.1 Newton's Law of Universal Gravitation What keeps the Moon in orbit around the Earth, or the planets circling the Sun? Isaac Newton answered this with one of the most important ideas in all of science: the **Law of Universal Gravitation**. He realized that the same force that makes an apple fall from a tree is the one that holds the cosmos together. This section explains this simple but powerful rule, showing how every object in the universe attracts every other object with a force that depends on their mass and the distance between them. ## Newton's Law of Universal Gravitation - States that *every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres*. - This law is fundamental to understanding the motion of celestial bodies (planets, moons, stars) and laid the foundation for classical mechanics. ## Mathematical Formulation For two spherical bodies of masses $m_1$ and $m_2$ separated by distance $r$, the gravitational force $F_g$ is given by: $$F_g = G \frac{m_1 \times m_2}{r^2}$$ Where: - $F_g$ is the gravitational force - $m_1$, $m_2$ are the masses of the two objects - $r$ is the distance between their centres - $G$ is the gravitational constant ## Gravitational Constant ($G$) A constant of proportionality in Newton's law of universal gravitation. Its value is $6.67 \times 10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}$. It was first accurately measured by Henry Cavendish in 1798 using a torsion balance apparatus, 100 years after Newton's work. This measurement was crucial for determining the masses of Earth and other astronomical bodies. ## Properties of Gravitational Force 1. **Always Attractive:** Gravitational force always pulls objects towards each other. 2. **Independent of Medium:** The force does not depend on the medium between the masses. 3. **Inverse-Square Force:** The magnitude of the gravitational force decreases rapidly with distance. If the distance between objects increases by a factor of $n$, the force decreases by a factor of $n^2$. For example, doubling the distance reduces the force to one-fourth. ## Consistency with Newton's Third Law According to Newton's law, if mass $m_1$ exerts a force $F_{21}$ on mass $m_2$, then mass $m_2$ exerts an equal and opposite force $F_{12}$ on mass $m_1$. Mathematically, $F_{12} = -F_{21}$. These forces form an action-reaction pair, demonstrating consistency with Newton's third law of motion. ## Why Called "Universal" It is called a universal law because it applies to *every* object in the universe attracting *every other* object, regardless of their size, composition, or location. It describes a fundamental force that governs motion from celestial bodies to everyday objects, consistent throughout the cosmos. ## Real-World Implications Despite being universal, the gravitational force between everyday objects is extremely small and goes unnoticed without highly sensitive instruments. For example, the force between two $50\ \mathrm{kg}$ bodies $1\ \mathrm{m}$ apart is about $1.5 \times 10^{-7}\ \mathrm{N}$. ## Worked Example **Q:** The mass of Earth is $6 \times 10^{24}\ \mathrm{kg}$ and that of the Moon is $7.4 \times 10^{22}\ \mathrm{kg}$, with a distance of $3.84 \times 10^{5}\ \mathrm{km}$ between them. Calculate the force exerted by Earth on the Moon. **A:** Using Newton's law of universal gravitation: $$F_g = G \frac{m_1 m_2}{r^2}$$ Given: - $m_{Earth} = 6 \times 10^{24}\ \mathrm{kg}$ - $m_{Moon} = 7.4 \times 10^{22}\ \mathrm{kg}$ - $r = 3.84 \times 10^{5}\ \mathrm{km} = 3.84 \times 10^{8}\ \mathrm{m}$ - $G = 6.67 \times 10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}$ Substitute these values into the formula to find $F_g$. | Concept | Description / Value | | :------------------ | :--------------------------------------------------- | | **Law Statement** | Every object attracts every other object... | | **Formula** | $F_g = G \frac{m_1 m_2}{r^2}$ | | **Gravitational Constant ($G$)** | $6.67 \times 10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}$ | | **Force Nature** | Always attractive; Inverse-square; Medium-independent | | **Relation to Newton's 3rd Law** | Gravitational forces are action-reaction pairs ($F_{12} = -F_{21}$) |