Two particles of mass 0.5 kg each go round a circle of radius 20 cm on accound of their mutual gravitational attraction. What will be the speed of each particle ?

Answer

To find the speed of each particle in a circular motion due to their mutual gravitational attraction, we can follow these steps:

### Step 1: Understand the System
We have two particles, each with a mass \( m = 0.5 \, \text{kg} \), moving in a circular path of radius \( r = 20 \, \text{cm} = 0.2 \, \text{m} \). The gravitational force between them provides the necessary centripetal force for their circular motion.

### Step 2: Identify the Forces
The gravitational force \( F_g \) between the two particles can be expressed using Newton's law of gravitation:
\[ F_g = \frac{G m_1 m_2}{d^2} \]
where \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \) is the gravitational constant, and \( d \) is the distance between the two particles.

### Step 3: Calculate the Distance Between the Particles
Since both particles are at a distance of \( r \) from the center of mass (which is at the midpoint between them), the total distance \( d \) between the two particles is:
\[ d = 2r = 2 \times 0.2 \, \text{m} = 0.4 \, \text{m} \]

### Step 4: Set Up the Equation for Centripetal Force
The centripetal force \( F_c \) required to keep each particle in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the speed of the particle.

### Step 5: Equate Gravitational Force and Centripetal Force
Since the gravitational force provides the centripetal force, we can set them equal:
\[ \frac{G m^2}{(2r)^2} = \frac{mv^2}{r} \]

### Step 6: Simplify the Equation
Substituting \( d = 2r \) into the gravitational force equation:
\[ \frac{G m^2}{4r^2} = \frac{mv^2}{r} \]
Now, we can cancel one \( m \) from both sides (since \( m \neq 0 \)):
\[ \frac{G m}{4r^2} = \frac{v^2}{r} \]

### Step 7: Rearrange to Solve for \( v^2 \)
Multiplying both sides by \( r \):
\[ \frac{G m}{4r} = v^2 \]

### Step 8: Solve for \( v \)
Taking the square root gives us:
\[ v = \sqrt{\frac{G m}{4r}} \]

### Step 9: Substitute the Values
Now we substitute the known values into the equation:
- \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( m = 0.5 \, \text{kg} \)
- \( r = 0.2 \, \text{m} \)

\[ v = \sqrt{\frac{6.67 \times 10^{-11} \times 0.5}{4 \times 0.2}} \]

### Step 10: Calculate the Speed
Calculating the above expression:
\[ v = \sqrt{\frac{6.67 \times 10^{-11} \times 0.5}{0.8}} = \sqrt{\frac{3.335 \times 10^{-11}}{0.8}} = \sqrt{4.16875 \times 10^{-11}} \approx 6.45 \times 10^{-6} \, \text{m/s} \]

Thus, the speed of each particle is approximately \( 6.45 \times 10^{-6} \, \text{m/s} \).