The moment of inertia of thin uniform circular disc about one of the diameter is I. Its moment of inertia about an axis perpendicular to the circular surface and passing through its centre is

Options

A. `(2l)/(3)`
B. 2l
C. `(l)/(2)`
D. 6l

Answer

To solve the problem of finding the moment of inertia of a thin uniform circular disc about an axis perpendicular to its surface and passing through its center, we can follow these steps:

### Step 1: Understand the Given Information
We know that the moment of inertia of the circular disc about one of its diameters is given as \( I \). This means that if we consider the disc lying in the xy-plane, the moment of inertia about the x-axis (one of the diameters) is \( I_x = I \).

### Step 2: Identify the Required Moment of Inertia
We need to find the moment of inertia of the disc about an axis that is perpendicular to its surface and passes through its center. This axis can be considered as the z-axis, and we denote its moment of inertia as \( I_z \).

### Step 3: Apply the Perpendicular Axis Theorem
The Perpendicular Axis Theorem states that for a planar body (like our circular disc), the moment of inertia about an axis perpendicular to the plane of the body (z-axis) is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane (x and y axes). Mathematically, this can be expressed as:
\[ I_z = I_x + I_y \]
where \( I_x \) and \( I_y \) are the moments of inertia about the x-axis and y-axis, respectively.

### Step 4: Recognize Symmetry
For a uniform circular disc, the moments of inertia about both diameters (x and y axes) are equal due to symmetry. Therefore, we can say:
\[ I_x = I_y = I \]

### Step 5: Substitute into the Perpendicular Axis Theorem
Now, substituting \( I_x \) and \( I_y \) into the equation from Step 3, we have:
\[ I_z = I + I = 2I \]

### Step 6: Conclusion
Thus, the moment of inertia of the thin uniform circular disc about the axis perpendicular to its surface and passing through its center is:
\[ I_z = 2I \]

### Final Answer
The moment of inertia about the axis perpendicular to the circular surface and passing through its center is \( 2I \).

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