7.1.1 Gravitational Force

According to legend, it was when an apple fell on Isaac Newton’s head that he had this epiphany: every mass attracts every other mass in the universe gravitationally.

$\displaystyle {{F}_{g}}=G\frac{{{{M}_{1}}{{M}_{2}}}}{{{{d}^{2}}}}$

More specifically, Newton’s Law of Gravitational states that the force of attraction F_g between any two point masses M₁ and M₂ is directly proportional to the product of the masses (M₁M₂) and inversely proportional to the square of the separation between them (d²).

G, the gravitational constant, has turned out to be a very small number. $G=6.67\times {{10}^{{-11}}}\text{ N k}{{\text{g}}^{{-2}}}\text{ }{{\text{m}}^{2}}$ .

Two things to note:

Non-point masses can often be treated as point masses at their centres of masses. So d is going to be the centre-to-centre distance.
The mutual force of attraction is an action-reaction pair. As dictated by N3L, two masses must always exert the same magnitude of gravitational pull on each other, even if they are of different masses.

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Example

Calculate the gravitational force of attraction between the two bowling balls shown below.

Solution

$\displaystyle \begin{aligned}{{F}_{g}}&=G\frac{{{{M}_{1}}{{M}_{2}}}}{{{{d}^{2}}}}\\&=(6.67\times {{10}^{{-11}}})\frac{{3.6\times 7.2}}{{{{{0.22}}^{2}}}}\\&=3.57\times {{10}^{{-8}}}\text{ N}\end{aligned}$