Two particles each of mass m seperated by a distance d and move in a uniform circle under the action of their mutual force of attraction. The speed of each particle is

Options

A. `sqrt((GM)/d)`
B. `sqrt((GM)/(4d))`
C. `sqrt((Gm)/(2d))`
D. `sqrt((2Gm)/d)`

Answer

To find the speed of each particle moving in a circular path under the action of their mutual gravitational attraction, we can follow these steps:

### Step 1: Understand the System
We have two particles, each of mass \( m \), separated by a distance \( d \). They are moving in a circular path due to their mutual gravitational attraction.

### Step 2: Identify the Forces
Each particle experiences two forces:
1. The gravitational force of attraction between the two particles.
2. The centripetal force required to keep each particle moving in a circular path.

### Step 3: Determine the Radius of Circular Motion
Since the two particles are separated by a distance \( d \), the radius \( r \) of the circular path for each particle is half of that distance:
\[ r = \frac{d}{2} \]

### Step 4: Write the Expression for Centripetal Force
The centripetal force \( F_c \) required to keep a particle of mass \( m \) moving in a circle of radius \( r \) with speed \( v \) is given by:
\[ F_c = \frac{mv^2}{r} \]
Substituting \( r = \frac{d}{2} \):
\[ F_c = \frac{mv^2}{\frac{d}{2}} = \frac{2mv^2}{d} \]

### Step 5: Write the Expression for Gravitational Force
The gravitational force \( F_g \) between the two particles is given by Newton's law of gravitation:
\[ F_g = \frac{G m m}{d^2} = \frac{G m^2}{d^2} \]

### Step 6: Set the Centripetal Force Equal to Gravitational Force
For the particles to maintain circular motion, the centripetal force must equal the gravitational force:
\[ \frac{2mv^2}{d} = \frac{G m^2}{d^2} \]

### Step 7: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{2v^2}{d} = \frac{G m}{d^2} \]
Now, multiply both sides by \( d \):
\[ 2v^2 = \frac{G m}{d} \]

### Step 8: Solve for \( v^2 \)
Dividing both sides by 2 gives:
\[ v^2 = \frac{G m}{2d} \]

### Step 9: Take the Square Root to Find \( v \)
Taking the square root of both sides, we find:
\[ v = \sqrt{\frac{G m}{2d}} \]

### Final Answer
The speed of each particle is:
\[ v = \sqrt{\frac{G m}{2d}} \]

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