Four particles each of mass and equidistant from each other, move along a circle of radius under the acting of their mutual gravitational attraction. The speed of each particle is:
Step 1: Given data
Four particles each of mass and equidistant from each other are move along a circle of radius under the acting of their mutual gravitational attraction.
Step 2: Formula used
Pythagoras theorem,
Gravitational force,
Centripetal force,
Step 3: Drawing a figure and calculating the distance between two masses
From the figure, by applying Pythagoras theorem, we get
Since the particles are arranged at equal distances symmetrically, the force on all the particles are same
Step 4: Finding gravitational force on each particle
From the figure, it is clear that the component of in the direction of are
Gravitational force on particle ,
Since, here AB and AC are same (B and C are symmetrical points)
Gravitational force on each of the four particle = gravitational force on particle A
Gravitational force on each of the four particle,
Step 5: Equating centripetal force and gravitational force to find the velocity of the particle
Since the particles also traverse on a circular path, they also experience a centripetal force towards the center. This centripetal force is balanced by the gravitational force experienced by the particle due to other particles.
Therefore,
Therefore, we get the velocity of each particle as .
Hence, Option B is correct.