Question

Two particles of equal mass ‘m’ move in a circle of radius ‘r’ under the action of their mutual gravitational attraction. The speed of each particle will be :

• A. \( \sqrt{\frac{GM}{2r}} \)
• B. \( \sqrt{\frac{GM}{4r}} \)
• C. \( \sqrt{\frac{GM}{r}} \)
• D. \( \sqrt{\frac{4GM}{r}} \)

Hint:

For orbital motion:

• Use gravitational force as the source of centripetal force.
• Solve for velocity using \( F_g = F_c \).

Answer 1

The Correct Option is B

The problem involves two particles of equal mass m moving in a circle of radius r due to their mutual gravitational attraction. We need to find the speed of each particle.

To solve this, we can use the concepts of centripetal force and Newton's law of gravitation.

Step 1: Identify Forces

The gravitational force between the two masses acts as the centripetal force required to keep each particle moving in a circular path. According to Newton's law of gravitation, the force between the two particles is given by:

F = \frac{Gm^2}{(2r)^2}

where G is the gravitational constant. Here, the distance between the masses is 2r because they are opposite each other on the circle.

Step 2: Apply the Concept of Centripetal Force

The necessary centripetal force to keep a mass moving in a circle is given by:

F_c = \frac{mv^2}{r}

where v is the speed of each particle.

Setting the gravitational force equal to the centripetal force, we have:

\frac{Gm^2}{4r^2} = \frac{mv^2}{r}

Step 3: Solve for the Speed v

Canceling m and simplifying the equation:

\frac{Gm}{4r} = v^2

Solving for v, we get:

v = \sqrt{\frac{Gm}{4r}}

Hence, the correct answer is: \sqrt{\frac{Gm}{4r}}.