The frequency `f` of vibrations of a mass `m` suspended from a spring of spring constant `k` is given by `f = Cm^(x) k^(y)` , where `C` is a dimensionnless constant. The values of `x and y` are, respectively,

Options

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

Explanation

To solve the problem, we need to find the values of \( x \) and \( y \) in the equation for frequency \( f = C m^x k^y \), where \( C \) is a dimensionless constant, \( m \) is mass, and \( k \) is the spring constant.

### Step-by-Step Solution:

1. **Identify the dimensions of frequency \( f \)**:
The frequency \( f \) has dimensions of time inverse, which can be represented as:
\[ [f] = T^{-1} \]

2. **Identify the dimensions of mass \( m \)**:
The mass \( m \) has the dimension:
\[ [m] = M^1 \]

3. **Identify the dimensions of spring constant \( k \)**:
The spring constant \( k \) is defined as force per unit length. The dimension of force is given by:
\[ [\text{Force}] = [\text{mass}] \times [\text{acceleration}] = M^1 L^1 T^{-2} \]
Therefore, the dimension of spring constant \( k \) is:
\[ [k] = \frac{[F]}{[L]} = \frac{M^1 L^1 T^{-2}}{L^1} = M^1 T^{-2} \]

4. **Write the dimensions of the equation**:
Substituting the dimensions of \( m \) and \( k \) into the equation \( f = C m^x k^y \):
\[ [f] = [C] [m]^x [k]^y \]
Since \( C \) is dimensionless, we have:
\[ [f] = [m]^x [k]^y = (M^1)^x (M^1 T^{-2})^y = M^{x+y} T^{-2y} \]

5. **Set up the equations based on dimensions**:
Now we equate the dimensions from both sides:
- For mass \( M \):
\[ x + y = 0 \quad \text{(1)} \]
- For time \( T \):
\[ -2y = -1 \quad \text{(2)} \]

6. **Solve the equations**:
From equation (2), we can solve for \( y \):
\[ -2y = -1 \implies y = \frac{1}{2} \]
Now substitute \( y \) back into equation (1):
\[ x + \frac{1}{2} = 0 \implies x = -\frac{1}{2} \]

7. **Final values**:
Thus, the values of \( x \) and \( y \) are:
\[ x = -\frac{1}{2}, \quad y = \frac{1}{2} \]

### Conclusion:
The values of \( x \) and \( y \) are \( -\frac{1}{2} \) and \( \frac{1}{2} \), respectively.