Chapter 5, Problem 78GP

(a)

To determine

To find: That why stars don’t crashes.

Explanation of Solution

Introduction:

The force of mutual gravity is providing the centripetal force.

The mutual attractions of stars are providing the necessary centripetal force for rotation. That’s why they don’t crashes with each other.

When two stars rotate in a circular motion, the gravitational force between the two stars is measured by

attraction between the two stars. So the stars will not hit each other.

Conclusion:

Therefore we conclude that as they are revolving with calculated speed so they don’t crash.

(b)

To determine

To find: The mass of each star.

Answer to Problem 78GP

  $9.6\times {10}^{26}\text{ kg}$

Explanation of Solution

Given Data:

  $D=8\times {10}^{10}$ m

  $T=12.6$ year

Formula Used:

Gravitational force between stars each of mass m is given by

  $F=G\frac{mm}{r^2}$

Speed v of star related to period of revolution T,

  $\begin{array}{cc}v& =\frac{2\pi (r/2)}{T}\\ & =\frac{\pi r}{T}\end{array}$

Centripetal force acts on each star,

  $F_{\text{c}}=\frac{m{v}^2}{(r/2)}$

Putting value of v in above equation

  $\begin{array}{cc}{F}_{\text{c}}& =\frac{m{\left(\frac{\pi r}{T}\right)}^2}{(r/2)}\\ & =\frac{2m{\pi }^{2}r}{T^2}\end{array}$

The centripetal force acting on the star is balanced by gravitational force between two stars

  $\begin{array}{cc}F& =F_{\text{c}}\\ G\frac{mm}{r^2}& =\frac{2m{\pi }^{2}r}{T^2}\\ m& =\frac{2{\pi }^2{r}^3}{G{T}^2}\end{array}$

Calculation:

Substituting the given values

  $\begin{array}{l}m=\frac{2{\pi }^2{r}^3}{G{T}^2}\\ =\frac{2{\pi }^2\times {\left(8.0\times {10}^{10}\text{m}\right)}^3}{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2\right)[{\left(\begin{array}{l}(12.6\text{y})\left(\frac{365.5\text{ days }}{1.0\text{ year }}\right)\hfill \\ \left(\frac{24\text{h}}{1.0\text{ day }}\right)\left(\frac{60\text{min}}{1.0\text{h}}\right)\left(\frac{60\text{s}}{1.0\text{min}}\right)\hfill \end{array}\right]}^2}\\ =9.6\times {10}^{26}\text{ kg}\end{array}$

Conclusion:

Therefore, the mass of stars are $9.6\times {10}^{26}\text{ kg}$ .