Question

The formula for critical velocity is:
A.) $\sqrt{2Rg}$
B.) ${\displaystyle \frac{R}{g}}$
C.) ${\displaystyle \frac{3R}{2g}}$
D.) $\sqrt{Rg}$

Answer 1

Hint: Study about the centripetal force and the gravitational force. Study how the satellite or any object moving in a circular direction around another object works. Think about which critical velocity you are asked by looking at the equations given as options.
Formula used:

$g={\displaystyle \frac{GM}{R^2}}$
${V_c}^2={\displaystyle \frac{GM}{R}}$

Complete step by step answer:
To put a satellite into a stable orbit around earth we gave them a constant horizontal velocity. The minimum velocity required to make them orbit in the stable orbit is called the critical velocity.
Consider an object of mass m orbiting another object of mass M. For the circular motion we need a centripetal force. In this case we have the necessary centripetal force due to the gravitational attraction.

Let the orbiting object is at a distance R from the object in the centre.

Now, for the motion to be stable the centripetal force should be equal to the gravitational attraction.

$\begin{array}{rl}& \text{centripetal force = gravitational force}\\ & {\displaystyle \frac{m{V_c}^2}{R}}=G{\displaystyle \frac{Mm}{R^2}}\\ & {V_c}^2={\displaystyle \frac{GM}{R}}\end{array}$

Where, G is the gravitational constant and $V_c$ is the critical velocity of the object.
Now, we can express the acceleration due to gravity as

$\begin{array}{rl}& g={\displaystyle \frac{GM}{R^2}}\\ & \text{so,}\\ & G={\displaystyle \frac{g{R}^2}{M}}\end{array}$

Now, putting the value of G in terms of g we get that,

$\begin{array}{rl}& {V_c}^2={\displaystyle \frac{GM}{R}}\\ & {V_c}^2={\displaystyle \frac{g{R}^2}{M}}{\displaystyle \frac{M}{R}}\\ & {V_c}^2=gR\\ & V_c=\sqrt{gR}\end{array}$

So, the correct option is (D)

Note: Critical velocity can also be defined as the maximum velocity with which a liquid can flow through a tube without being turbulent.

$V_c={\displaystyle \frac{R_e\eta }{\rho r}}$

Where, $R_e$is the Reynolds number, $\eta$ is the viscosity, $\rho$is the density of the fluid and r is the radius of the tube.

Do not confuse this critical velocity of fluid with the above critical velocity for circular motion.