The period `T` of a soap bubble under `SHM` is given by `T = P^(a) D^(b) S^(c )`, where `P` is pressure, `D`, is density and `S` is surface tension. Then the values of `a,b` and `c` are

Options

A. `- (3)/(2), (1)/(2), 1`
B. `-1, -2, 3`
C. `(1)/(3), - (3)/(2), - 1//2`
D. `1, 2, (3)/(2)`

Explanation

To solve the problem of finding the values of \( a \), \( b \), and \( c \) in the equation \( T = P^a D^b S^c \), where \( T \) is the period of a soap bubble under simple harmonic motion (SHM), we will use dimensional analysis.

### Step-by-Step Solution:

1. **Identify the Dimensions**:
- The dimension of time period \( T \) is given by:
\[ [T] = M^0 L^0 T^1 \]
- The dimension of pressure \( P \) is defined as force per unit area:
\[ [P] = \frac{F}{A} = \frac{M L T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]
- The dimension of density \( D \) is mass per unit volume:
\[ [D] = \frac{M}{L^3} = M^1 L^{-3} T^0 \]
- The dimension of surface tension \( S \) is force per unit length:
\[ [S] = \frac{F}{L} = \frac{M L T^{-2}}{L} = M^1 L^0 T^{-2} \]

2. **Set Up the Dimensional Equation**:
- The equation relating the dimensions is:
\[ [T] = [P]^a [D]^b [S]^c \]
- Substituting the dimensions we found:
\[ M^0 L^0 T^1 = (M^1 L^{-1} T^{-2})^a (M^1 L^{-3} T^0)^b (M^1 L^0 T^{-2})^c \]

3. **Expand the Right-Hand Side**:
- Expanding the right-hand side gives:
\[ M^{a+b+c} L^{-a-3b} T^{-2a-2c} \]

4. **Equate Dimensions**:
- Now, we equate the powers of \( M \), \( L \), and \( T \) from both sides:
- For \( M \):
\[ a + b + c = 0 \quad \text{(1)} \]
- For \( L \):
\[ -a - 3b = 0 \quad \text{(2)} \]
- For \( T \):
\[ -2a - 2c = 1 \quad \text{(3)} \]

5. **Solve the Equations**:
- From equation (2):
\[ a = -3b \quad \text{(4)} \]
- Substitute (4) into (1):
\[ -3b + b + c = 0 \implies -2b + c = 0 \implies c = 2b \quad \text{(5)} \]
- Substitute (4) and (5) into (3):
\[ -2(-3b) - 2(2b) = 1 \implies 6b - 4b = 1 \implies 2b = 1 \implies b = \frac{1}{2} \]
- Substitute \( b \) back into (4) and (5):
\[ a = -3 \left(\frac{1}{2}\right) = -\frac{3}{2} \]
\[ c = 2 \left(\frac{1}{2}\right) = 1 \]

6. **Final Values**:
- The values of \( a \), \( b \), and \( c \) are:
\[ a = -\frac{3}{2}, \quad b = \frac{1}{2}, \quad c = 1 \]

### Summary:
The values of \( a \), \( b \), and \( c \) are:
- \( a = -\frac{3}{2} \)
- \( b = \frac{1}{2} \)
- \( c = 1 \)