 # Law of Gravitation

## Newton's Law of Gravity

\$\$F = G⋅{M_1M_2}/{r^2}\$\$

where G is Newton's constant of universal gravity: \$G = 6.667 ⋅ 10^{-11} N m^{-2} kg^{-2}\$, \$M_1\$ and \$M_2\$ are the masses of two objects separated by distance r.

Newton's brilliant insight into the nature of gravity was that it is a force acting at a distance, and it is the same force which draws an apple to the Earth and holds the moon in orbit. Newton's Law assumes point masses - that is mathematically placing all the mass of a planet at its centre as a single, dimensionless point. A problem with this equation is that as the distance, r, between the masses approaches zero, the force approaches infinity.

This can also be expressed as the gravitational field near the Earth's surface: \$9.81 N/{kg}\$.

## Gravitational Field

Assuming a point source of gravity, the force of gravitational attraction on the surface of the Earth, mass \$M_e\$, on a smaller mass, \$m\$, is the weight of the object, mg:

Hence, \$mg = G{M_em}/{R_e^2}\$ ⇒ \$g = {GM_e}/{R_e^2}\$

\$\$g = G⋅{M}/{r^2}\$\$

A gravitational field at a certain point is the force per unit mass experienced by a small mass at that point. The unit of gravitational field strength is \$N kg^{-1}\$.

### Mass and Weight

Although our bathroom scales insist on giving us the bad news in kilograms, they cannot measure our mass directly. What they are measuring is the force of attraction between your mass and the Earth's mass.

To be more precise, the meter of the scales is indicating the degree to which the spring inside the scales is compressing, which is due to the reaction force to the gravitational force, a consequence of Newton's Third Law of Motion.

Since we know fairly accurately the strength of the gravitational field, g, at the Earth's surface, we can determine the mass (kg) by dividing the force (in newtons N) by g (\$9.81 m s^{-2}\$. The scales should therefore be giving your weight in newtons.

The Earth is not a perfect sphere, and there is a variation in crustal density around the globe, so the value for g varies depending on where you are. This means that your weight is location specific. Your mass will be the same in all measurements, and is still the same mass even if you are in microgravity (in orbit or in free space) or in freefall. Under these circumstances, you are not pushing against anything, so do not feel any reaction force, which is the sensation of weight.

## Gravitational Potential Energy The gravitational potential is a measure of the work done to move a unit mass from infinity, where the potential is zero, to a point where the potential is not zero. Since it requires positive work to move a mass further away from another mass, allowing a mass to approach another mass requires negative work.

The gravitational potential V is the ratio of the work W done on a mass m in moving it from infinity to a point P:

\$\$V = W/m\$\$

The gravitational potential due to a single mass M a distance r from the centre of M is:

\$\$V = -{GM}/r\$\$

The unit of V is \$J kg^{-1}\$.

A mass m at a point in space with gravitational potential V is:

\$\$E_p = mV\$\$

the work done to move a mass m from one gravitational potential point to another is:

\$\$W = m(V_2 - V_1) = mΔV\$\$

## Escape Velocity

The total energy of a mass m (you and your rocketship) moving at speed v, distance r from the centre of large mass M (a planet) is:

\$\$E = 1/2mv^2 - {GMm}/r\$\$

What is the speed necessary for mass m to be able to reach infinity?

At infinity, the potential energy is zero, so: \$1/2mv^2 - {GMm}/r =0\$, where v is the smallest velocity which allows the mass to reach infinity. This is the escape velocity:

\$\$v_{escape} = √{2GM}/r\$\$

Related scientists: Einstein, Hawking, Randall, Laplace, Maxwell.

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