A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and z' axes will be :

Options

A. `1 : 5`
B. ` 1 : 3`
C. `1 : 4`
D. ` 1: 2`

Answer

To find the ratio of the moment of inertia of a thin circular disk about the z-axis and the z' axis, we can follow these steps:

### Step 1: Moment of Inertia about the z-axis
The moment of inertia (I_z) of a thin circular disk about its central axis (z-axis) is given by the formula:
\[ I_z = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disk and \( r \) is the radius of the disk.

### Step 2: Moment of Inertia about the z' axis
To find the moment of inertia about the z' axis (which is parallel to the z-axis and located at a distance \( r \) from it), we can use the parallel axis theorem. The parallel axis theorem states that:
\[ I_{z'} = I_z + m d^2 \]
where \( d \) is the distance between the two axes. In this case, \( d = r \).

Substituting the values, we get:
\[ I_{z'} = \frac{1}{2} m r^2 + m r^2 \]
\[ I_{z'} = \frac{1}{2} m r^2 + \frac{2}{2} m r^2 = \frac{3}{2} m r^2 \]

### Step 3: Ratio of Moments of Inertia
Now, we can find the ratio of the moment of inertia about the z-axis to that about the z' axis:
\[ \text{Ratio} = \frac{I_z}{I_{z'}} = \frac{\frac{1}{2} m r^2}{\frac{3}{2} m r^2} \]
The \( m r^2 \) terms cancel out:
\[ \text{Ratio} = \frac{1/2}{3/2} = \frac{1}{3} \]

### Conclusion
Thus, the ratio of the moment of inertia about the z-axis to that about the z' axis is:
\[ \text{Ratio} = 1 : 3 \]