Physics with Calculus/Mechanics/Newton's Law of Gravitation and Weight

Newton's Law of Universal Gravitation

As remarked previously, gravitation is one of the four classes of interactions found in nature, and it was the earliest of the four to be studied extensively. Isaac Newton discovered in the 17th century that the same interaction that makes an apple fall from a tree also keeps planets in orbit around the sun. Along with his three laws of motion, Newton published the law of gravitation in 1687. It can be stated as follows:

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

In mathematical terms, the law of universal gravitation may be given by $F=G{\frac {m_{1}m_{2}}{r^{2}}},$

where:

• F is the magnitude of the gravitational force between the two point masses,
• G is the gravitational constant or $G=\left(6.67428\pm 0.00067\right)\times 10^{-11}\ {\mbox{m}}^{3}\ {\mbox{kg}}^{-1}\ {\mbox{s}}^{-2}$,
• m₁ is the mass of the first point mass,
• m₂ is the mass of the second point mass,
• r is the distance between the two point masses.

Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.

$\mathbf {F} _{12}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{12}\vert }^{2}}\,\mathbf {\hat {r}} _{12}$

where

$\mathbf {F} _{12}$ is the force applied on object 2 due to object 1

$G$ is the gravitational constant

$m_{1}$ and $m_{2}$ are respectively the masses of objects 1 and 2

$\vert \mathbf {r} _{12}\vert \ =\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert$ is the distance between objects 1 and 2

$\mathbf {\hat {r}} _{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}$ is the unit vector from object 1 to 2

It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F₁₂ = − F₂₁.

Gravitational acceleration is the acceleration of an object caused by the force of gravity from another object. In the absence of any other forces, any object will accelerate in a gravitational field at the same rate, regardless of the mass of the object. On the surface of the Earth, all objects fall with an acceleration of somewhere between 9.78 and 9.82 m/s² depending on latitude, with a conventional standard value of exactly 9.80665 m/s², (approx. 32.174 ft/s²).

The gravitational acceleration towards an object is given by:

$\mathbf {g} =-{mG \over r^{2}}\mathbf {\hat {r}}$

where:

m is the mass of the object,

r is the distance from center of the object to the location we are considering,

$\mathbf {\hat {r}}$ is the unit length vector from center of the object to the location we are considering,

G is the gravitational constant of the universe.

As one can see, the gravitational acceleration is independent of the mass of the reference particle.

The weight of a body is the total gravitational force exerted on the body by all other bodies in the universe. When the body is near the surface of the earth, one may neglect all other gravitational forces and consider the weight as just the earth's gravitational attraction. If we model the earth as a spherically symmetric body, then the weight of a small body with mass m at the earth's surface is

$w=F=G{\frac {m_{1}m_{2}}{r^{2}}}$