For torsional oscillator, `sqrt(I//C)` has dimensions of ? Where `C` is torsional constant and `I` is moment of inertia :

Options

A. `M^(0)L^(0)T^(-1)`
B. `M^(0)L^(0)T^(1)`
C. `M^(0)L^(1)T^(-1)`
D. `M^(0)L^(0)T^(2)`

Answer

To find the dimensions of \(\sqrt{\frac{I}{C}}\) for a torsional oscillator, we will break down the problem into steps.

### Step 1: Identify the dimensions of moment of inertia \(I\)
The moment of inertia \(I\) is defined as:
\[ I = m \cdot r^2 \]
where \(m\) is mass and \(r\) is the radius (or distance from the axis of rotation).

The dimensions of \(I\) can be expressed as:
\[ \text{Dimension of } I = [M][L^2] = M L^2 \]

### Step 2: Identify the dimensions of the torsional constant \(C\)
The torsional constant \(C\) is defined as the ratio of torque \(\tau\) to the angle of twist \(\phi\):
\[ C = \frac{\tau}{\phi} \]
Since \(\phi\) (angle of twist) is dimensionless, the dimensions of \(C\) are the same as the dimensions of torque \(\tau\).

Torque is defined as:
\[ \tau = \text{Force} \times \text{Distance} \]
The dimension of force is given by:
\[ \text{Force} = [M][L][T^{-2}] \]
Thus, the dimension of torque is:
\[ \text{Dimension of } \tau = [M][L][T^{-2}] \cdot [L] = M L^2 T^{-2} \]
Therefore, the dimensions of \(C\) are:
\[ \text{Dimension of } C = M L^2 T^{-2} \]

### Step 3: Calculate the dimensions of \(\frac{I}{C}\)
Now we can find the dimensions of \(\frac{I}{C}\):
\[ \frac{I}{C} = \frac{M L^2}{M L^2 T^{-2}} = \frac{M L^2}{M L^2} \cdot T^{2} = T^{2} \]

### Step 4: Find the dimensions of \(\sqrt{\frac{I}{C}}\)
Taking the square root of the dimensions we found:
\[ \sqrt{\frac{I}{C}} = \sqrt{T^{2}} = T \]

### Final Answer
Thus, the dimensions of \(\sqrt{\frac{I}{C}}\) are:
\[ \text{Dimension of } \sqrt{\frac{I}{C}} = T \]