7.1.2 Satellite Motion

Because of the small value of G, ( $G=6.67\times {{10}^{{-11}}}\text{ N k}{{\text{g}}^{{-2}}}\text{ }{{\text{m}}^{2}}$ ), gravitational force is a negligibly weak force between everyday objects of ordinary masses. However, the gravitational force exerted by massive bodies such as planets and stars are large and influential. Gravitation is the reason why apples fall to the ground, satellites orbit the Earth and planets orbit the Sun.

Example

Ignoring air resistance, calculate the required speed v₀ for an object to go into orbit just above the Earth’s surface.

Earth’s radius $=6370\text{ km}$

Earth’s mass $=5.97\times {{10}^{{24}}}\text{ kg}$

Solution

Gravitational pull provides the required centripetal force.

$\displaystyle \begin{aligned}{{F}_{g}}&={{F}_{c}}\\G\frac{{Mm}}{{{{r}^{2}}}}&=m\frac{{{{v}^{2}}}}{r}\\v&=\sqrt{{\frac{{GM}}{r}}}\\&=\sqrt{{\frac{{(6.67\times {{{10}}^{{-11}}})(5.97\times {{{10}}^{{24}}})}}{{6370\times {{{10}}^{3}}}}}}\\&=7910\text{ m }{{\text{s}}^{{-1}}}\end{aligned}$