Newtons Laws EX 42

The gravitational force is the only force that acts on the object of mass $m$ on the surface of a neutron star of mass $M_S$. Note that in this calculation we will use the Newton's Law of gravitation although it may be only approximately valid to use it. The object on the surface of this star has radial acceleration and so we need to determine the speed of this object and then compute the acceleration:

$v=\frac{2\pi r}{T}$

$a_r=\frac{v^2}{r}=\frac{(\frac{2\pi r}{T}{)}^2}{r}=\frac{4{\pi }^{2}r}{T^2}$

The gravitational attraction by the neutron star is the force that causes this acceleration and thus, we can write Newton's 2nd Law:

$\frac{G{M}_{S}m}{r^2}=m{a}_r=m\frac{4{\pi }^{2}r}{T^2}$

By solving for $M_S$ and then by substituting the values for the gravitational constant, the radius of the neutron star and the period of rotation into this equation, we find the mass of the neutron star to be

$M_S=\frac{4{\pi }^2{r}^3}{G{T}^2}=\frac{4{\pi }^2(20\times 10^3{)}^3}{(6.67\times 10^{-11})(1{)}^2}=4.7\times 10^{24}kg$

Note the awesome mass of this neutron star with a radius of only 20 km!