_{Our Sun is${\mathbf{\text{2.3×10}}}^{\mathbf{\text{4}}}\mathbf{\text{ly}}$ (light-years) from the center of our Milky Way galaxy and is moving in a circle around that center at a speed of$\mathbf{\text{250\hspace{0.17em}km/s}}$ . (a) How long does it take the Sun to make one revolution about the galactic center? (b) How many revolutions has the Sun completed since it was formed about ${\mathbf{\text{4.5×10}}}^{\mathbf{\text{9}}}$years ago?}

Use the relation between velocity and time period.

Formula:

i)$v=r\omega =2\pi r/T$

ii)$N=\frac{\text{totaltime}}{\text{timeforonerevolution}}$

Where,

v is velocity, r is radius, T is time, N is number of revolutions and $\omega$ is angular velocity

First, convert$2.3\times {10}^4\text{\hspace{0.17em}ly}$into km as follows:

$r=2.3\times {10}^4\text{ly}\times \frac{9.46\times {10}^{12}\text{\hspace{0.17em}km}}{1\text{ly}}=2.18\times {10}^{17}\text{km}$

Now,

$v=r\omega =2\pi r/T$

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

$t=\frac{2\pi (2.18\times {10}^{17})}{250}$

$t=5.5\times {10}^{15}\text{sec}$

Hence, it takes $5.5\times {10}^{15}\text{\hspace{0.17em}sec}$ for the sun to make one revolution about the galactic center.

Now, number of revolution as follows:

First, convert time$4.5\times {10}^9\text{year}$into sec,

$4.5\times {10}^9\times \frac{\text{365\hspace{0.17em}days}}{\text{1year}}\times \frac{24\text{hour}}{1\text{\hspace{0.17em}day}}\times \frac{3600\text{\hspace{0.17em}sec}}{1\text{\hspace{0.17em}hour}}=1.422\times {10}^{17}\text{sec}$

Now, number of revolutions are,

$N=\frac{\text{totaltime}}{\text{timeforonerevolution}}$

$N=\frac{1.422\times {10}^{17}}{5.5\times {10}^{15}}=25.8$

In two significant figures,

$N=26\text{revolutions}$

Hence,$26$revolutions have been completed since the sun was found about$4.5\times {10}^9$years ago.

Therefore, using the formula for velocity, the time can be found. Using this time, the number of revolutions of the sun can be found.