Animated Solution for Physics - Gravitation: Two identical particles of mass 1 kg each go round a circle of radius $R$, under the action of their mutual gravitational attraction. The angular speed of each particle is

Binary Systems

Imagine looking up at the night sky and observing two identical stars locked in a perpetual cosmic dance. They orbit a common center, never colliding, yet never flying apart. This beautiful phenomenon is exactly what our problem models: two identical particles of mass $m$ moving in a perfect circle of radius $R$.

For these two particles to maintain this stable circular orbit, they must always be diametrically opposite to each other. If they weren't, the gravitational force would pull them off their circular path. Because they are on opposite ends of the circle, the distance separating them is exactly the diameter of the circle, which is $2R$.

What keeps these particles moving in a circle? According to Newton's First Law, an object will move in a straight line unless acted upon by a force. To move in a circle, there must be a force constantly pulling the particles toward the center. This is the centripetal force.

In our cosmic setup, there are no strings or invisible tracks. The only force acting on the particles is their mutual gravitational attraction. Therefore, the gravitational pull itself must provide the exact amount of centripetal force required to keep them in orbit.

Let's translate this physical intuition into mathematics. First, we write down Newton's Law of Universal Gravitation for the force between the two particles:

Substituting our specific values, where both masses are $m$ and the separation distance is $2R$:

Next, we express the required centripetal force for a particle of mass $m$ moving in a circle of radius $R$ with an angular speed $\omega$:

Since the gravitational force provides the centripetal force, we equate the two expressions:

Now, it's just a matter of algebra. We can cancel one mass $m$ from both sides of the equation:

Isolating ${\omega }^2$ by dividing both sides by $R$:

Taking the square root of both sides gives us the angular speed:

Finally, the problem states that the mass of each particle is exactly $1\text{ kg}$. Substituting $m=1$ into our expression yields the final answer:

This elegant result shows how the fundamental laws of gravitation and circular motion intertwine to govern the dynamics of orbiting bodies.