Question

Two asteroids of equal masses revolve diametrically opposite tho each other on circle of radius 1000km with equal velocity. If mass of one them is 1,00,00,00,000kg, then find their velocity.

Answer 1

Given :

Mass of each asteroid revolving diametrically opposite to each other = 1,00,00,00,000 kg

Radius of circle in which asteroids are revolving = 100 km

To Find ;

If they are moving with equal velocity then their velocity = ?

Solution :

We know that for a circular motion  under gravitational force :

Centrifugal force = gravitational force   -(1)

∴Centrifugal force is given as  = $\frac{mv^2}{r}$    -(2)

And gravitational force is given as = $\frac{Gm_1m_2}{r^2}$   -(3)

The distance between the two asteroids will be equal to diameter of the circle in which they are revolving i.e. 2R

So, using equation 1, 2 and 3 and putting the given values we get :

$\frac{mv^2}{R} = \frac{Gmm}{(2R)^2}$

Or, $\frac{mv^2}{R}= \frac{Gm^2}{4R^2}$

Or , $v = \sqrt{\frac{Gm}{4R}}$

Now , on putting the given numerical  values of radius and mass we get ;

$v = \sqrt{\frac{6.67 \times 10 ^{-11} \times 10^9 kg}{4 \times 1000\times 1000 m } }$

 $= \sqrt{\frac{6.67\times 10^{-8}}{4} }$

 $=\sqrt{6.67} \times \frac{10^{-4}}{2} \frac{m}{s}$

 $= \frac{2.53}{2}\times 10^{-4}\frac{m}{s}$

v = $1.29 \times 10^{-4}\frac{m}{s}$

So, the velocity of each asteroid is $1.29 \times 10^{-4}\frac{m}{s}$