Question

The mass density of a spherical galaxy$K$ varies$K/r$ over a large distance$\text{'}r\text{'}$ from its center. In that region, a small star is in a circular orbit of radius$R$. Then the period of a revolution$T$ depends on$R$ as:

• A. $T^2\propto R$
• B. $T^2\propto R^3$
• C. $T^2\propto {\left(1/R\right)}^3$
• D. $T\propto R$

Answer 1

The correct option is A

$T^2\propto R$

Step 1. Given data:

The mass density of a spherical galaxy$=K$

The distance from the center$=r$

The radius of the circular orbit$=R$

The period of a revolution$=T$

Step 2. Formula used:

Mass of the galaxy$={\int }_0^R\rho dV$
The force is due to angular acceleration $F=m{\omega }^{2}R$ ($\omega =$Angular velocity)

The force due to gravity $F=\frac{GMm}{R^2}$ ($G=$Gravitational constant)

Step 3. Determining the relation between$T,R$

The total mass of the galaxy,

$\begin{array}{rcl}M& =& {\int }_0^R\frac{k}{r}.4\pi r^{2}dr\\ & =& 2\pi k{R}^2\\ m{\omega }^{2}R& =& \frac{GMm}{R^2}\\ & \Rightarrow & {\omega }^2=\frac{2\pi k{R}^{2}G}{R^3}\\ & =& \frac{2\pi kG}{R}\end{array}$

As we know,

$\begin{array}{rcl}T& =& \frac{2\pi }{\omega }\\ & \Rightarrow & {\omega }^2\propto 1/R\\ & \Rightarrow & T^2\propto R\end{array}$

Thus, the period of a revolution$T$ depends on$R$ as$T^2\propto R$
Hence, option A is the correct answer.