A satellite is orbiting around a planet. Its orbital velocity `(V_(0))` is found to depend upon
(A) Radius of orbit (R)
(B) Mass of planet (M)
(C ) Universal gravitatin contant (G)
Using dimensional anlaysis find and expression relating orbital velocity `(V_(0))` to the above physical quantities.

Answer

To find an expression for the orbital velocity \( V_0 \) of a satellite in terms of the radius of orbit \( R \), the mass of the planet \( M \), and the universal gravitational constant \( G \), we will use dimensional analysis.

### Step 1: Identify the dimensions of the quantities involved

1. **Orbital Velocity \( V_0 \)**: The dimension of velocity is given by:
\[ [V_0] = [L][T^{-1}] = L T^{-1} \]
where \( L \) is length and \( T \) is time.

2. **Radius of Orbit \( R \)**: The dimension of radius is:
\[ [R] = [L] \]

3. **Mass of Planet \( M \)**: The dimension of mass is:
\[ [M] = [M] \]

4. **Universal Gravitational Constant \( G \)**: The dimension of \( G \) can be derived from the formula for gravitational force:
\[ F = \frac{G m_1 m_2}{R^2} \]
Rearranging gives:
\[ G = \frac{F R^2}{m_1 m_2} \]
The dimension of force \( F \) is:
\[ [F] = [M][L][T^{-2}] = M L T^{-2} \]
Thus, substituting into the equation for \( G \):
\[ [G] = \frac{M L T^{-2} \cdot L^2}{M^2} = M^{-1} L^3 T^{-2} \]

### Step 2: Formulate the relationship using dimensional analysis

Assume that the orbital velocity \( V_0 \) can be expressed as a function of \( R \), \( M \), and \( G \):
\[ V_0 = k R^a M^b G^c \]
where \( k \) is a dimensionless constant, and \( a \), \( b \), and \( c \) are the powers we need to determine.

### Step 3: Write the dimensions of the right-hand side

Substituting the dimensions we found:
\[ [V_0] = [R^a] [M^b] [G^c] = [L^a] [M^b] [M^{-1} L^3 T^{-2}]^c \]
This simplifies to:
\[ [V_0] = L^a M^b M^{-c} L^{3c} T^{-2c} = L^{a + 3c} M^{b - c} T^{-2c} \]

### Step 4: Set up equations based on dimensions

Since both sides must be dimensionally equal, we can equate the powers of \( L \), \( M \), and \( T \):
1. For \( L \):
\[ a + 3c = 1 \quad (1) \]
2. For \( M \):
\[ b - c = 0 \quad (2) \]
3. For \( T \):
\[ -2c = -1 \quad (3) \]

### Step 5: Solve the equations

From equation (3):
\[ c = \frac{1}{2} \]

Substituting \( c \) into equation (2):
\[ b - \frac{1}{2} = 0 \implies b = \frac{1}{2} \]

Substituting \( c \) into equation (1):
\[ a + 3 \times \frac{1}{2} = 1 \implies a + \frac{3}{2} = 1 \implies a = 1 - \frac{3}{2} = -\frac{1}{2} \]

### Step 6: Write the final expression

Now substituting the values of \( a \), \( b \), and \( c \) back into the expression for \( V_0 \):
\[ V_0 = k R^{-\frac{1}{2}} M^{\frac{1}{2}} G^{\frac{1}{2}} \]
This can be rewritten as:
\[ V_0 = k \frac{\sqrt{M G}}{\sqrt{R}} \]

### Conclusion

Thus, the expression relating the orbital velocity \( V_0 \) to the radius of orbit \( R \), mass of the planet \( M \), and universal gravitational constant \( G \) is:
\[ V_0 = k \sqrt{\frac{M G}{R}} \]