Final answer:
The time it takes for two particles with mass m to reduce their separation under mutual gravitational attraction is represented by π/√Gm. This result is derived using Newton's laws of gravitation and kinematics, considering the movement of two equal masses towards their common center of mass. Option B is correct.
Step-by-step explanation:
The question asks about the time it takes for two particles with mass m to reduce their separation from a distance a to a distance 2a under their mutual gravitational attraction.
Using Newton's law of universal gravitation, the initial acceleration of each particle can be found through the formula F = GmM / r², where F is the gravitational force, G is the gravitational constant (6.674 × 10⁻¹¹ N.m²/kg²), m and M are the masses of the two bodies (which are identical in this case), and r is the separation between the centers of the two masses. From Newton's second law, F can also be expressed as m x a, where a is the acceleration.
Using the symmetry of the problem, we know that both masses will move towards each other, meeting halfway (at a/2 from their starting points) due to their equal mass and the uniform gravitational force applied. This can also be treated as a two-body problem in which both particles experience the same acceleration due to the gravitational force between them.
By analyzing the kinematics of the particles under constant acceleration and using the appropriate kinematic equations, we are led to the conclusion that the time to half the distance between them is proportional to the inverse square root of the product of the gravitational constant G and the mass m.
The correct answer is therefore option b. π/√Gm, which represents the period of a small amplitude oscillation of the two-body system and is proportional to the inverse square root of the product of the gravitational constant and the particles' mass.