FHSST Physics/Newtonian Gravitation/Properties

Properties

Gravitational attraction is a force and therefore must be described by a vector - so remember magnitude and direction. The force due to gravity acts between any two objects with mass. To determine the magnitude of the force we use the following equation:

${\begin{matrix}F={\frac {Gm_{1}m_{2}}{r^{2}}}\end{matrix}}$

(9.1)

This equation describes the force between two bodies, one of mass m₁, the other of mass m₂ (both have units of kilograms, or kg for short). The G is Newton's`Gravitational Constant' (6.673×10⁻¹¹ N·m²·kg⁻²) and r is the straight line distance between the two bodies in meters.

This means the bigger the masses, the greater the force between them. Simply put, big things matter big with gravity. The 1/r² factor (or you may prefer to say r⁻²) tells us that the distance between the two bodies plays a role as well. The closer two bodies are, the stronger the gravitational force between them is. We feel the gravitational attraction of the Earth most at the surface since that is the closest we can get to it, but if we were in outer-space, we would barely even know the Earth's gravity existed!

Remember that

${\begin{matrix}F=ma\end{matrix}}$

(9.2)

which means that every object on the earth feels the same gravitational acceleration! That means whether you drop a pen or a book (from the same height), they will both take the same length of time to hit the ground... in fact they will be head to head for the entire fall if you drop them at the same time. We can show this easily by using the two equations above (9.1 and 9.2). The force between the Earth (which has the mass m_e) and an object of mass m_o is

${\begin{matrix}F={\frac {Gm_{o}m_{e}}{r^{2}}}\end{matrix}}$

(9.3)

and the acceleration of an object of mass m_o (in terms of the force acting on it) is

${\begin{matrix}a_{o}={\frac {F}{m_{o}}}\end{matrix}}$

(9.4)

So we substitute equation (9.3) into equation (9.4), and we find that

${\begin{matrix}a_{o}={\frac {Gm_{e}}{r^{2}}}\end{matrix}}$

(9.5)

Since it doesn't matter what m_o is, this tells us that the acceleration on a body (due to the Earth's gravity) does not depend on the mass of the body. Thus all objects feel the same gravitational acceleration. The force on different bodies will be different but the acceleration will be the same. Due to the fact that this acceleration caused by gravity is the same on all objects we label it differently, instead of using a we use g which we call the gravitational acceleration.