Problem 87 - A Complete Resource Book in Physics for JEE Main

Question

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is (A) \(\sqrt{\frac{G M}{R}}\) (B) \(\sqrt{2 \sqrt{2} \frac{G M}{R}}\) (C) \(\sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\) (D) \(\frac{1}{2} \sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\)

Step-by-Step Solution

Answer

The speed of each particle is: (A) \(\sqrt{\frac{G M}{R}}\)

Step 1: Calculate the gravitational force between a pair of particles

We will calculate the gravitational force between two particles at opposite ends of a diameter of the circle with the radial distance being 2R (half the length of the square's diagonal). To determine this force, we will use Newton's law of universal gravitation: \[F = G \frac{M_1M_2}{r^2}\] where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(M_1\) and \(M_2\) are the masses of the particles, and \(r\) is the radial distance between them. Since we have four particles, all with mass M, and the radial distance is 2R: \[F = G \frac{M \cdot M}{(2R)^2}\]

Step 2: Find the centripetal force on each particle

Now, we will determine the centripetal force acting on each particle. The centripetal force is responsible for keeping the particles moving in a circular path. The centripetal force formula is: \[F_c = \frac{mv^2}{r}\] where \(F_c\) is the centripetal force, \(m\) is the mass of the particle, \(v\) is the speed, and \(r\) is the radius of the circle the particle is moving around.

Step 3: Set the gravitational force equal to the centripetal force

The forces acting on the particles are in balance, so the gravitational force between a pair of particles must equal the centripetal force acting on each of them. Therefore: \[G \frac{M \cdot M}{(2R)^2} = \frac{Mv^2}{R}\]

Step 4: Solve for the speed of the particles

Now, we'll solve for the speed (v) of the particle in the equation: \[Mv^2 = \frac{G M^2}{4R}\] Divide both sides by M, and then take the square root of both sides: \[v = \sqrt{\frac{G M}{4R}}\]

Step 5: Compare the answer with the given options

Multiplying the speed expression by 2 to account for the two opposite particles, we get: \[2v = \sqrt{\frac{G M}{R}}\] Thus, the speed of each particle is: (A) \(\sqrt{\frac{G M}{R}}\) Comparing our answer with the given options, we find that option (A) is correct.