The moment of inertia of an uniform thin sheet of mass M of the given shape about the specified axis is (axis and sheet both are in same plane:)

To address the problem of finding the moment of inertia of a uniform thin sheet of mass $M$ about a specified axis, we will follow these steps:

1. Break Down the Problem

1. Identify the shape of the uniform thin sheet.
2. Determine the position of the axis about which the moment of inertia is to be calculated.
3. Establish the formula for the moment of inertia relevant to the shape.

2. Relevant Concepts

• The moment of inertia $I$ is calculated using the formula: $I = \int r^2 \, dm$ where $r$ is the perpendicular distance from the axis of rotation to the mass element $dm$.
• For common shapes, specific formulas are used. For instance:
  • For a rectangle about its center: $I = \frac{1}{12} M (b^2 + h^2)$
  • For a circular sheet about its center: $I = \frac{1}{2} M r^2$

3. Analysis and Detail

• For specific shapes, calculate $I$ by integrating across the geometry of the shape or using known formulas to simplify the calculation.
• For complex shapes, you may have to break them down into simpler shapes, calculate each moment of inertia individually, and then use the parallel axis theorem if necessary.

4. Verify and Summarize

• Confirm that the area and mass distribution correspond appropriately to the sheet considered.
• Ensure that the correct axis and distance $r$ from the axis to the mass element are being used.

Final Answer

The final expression for the moment of inertia $I$ will depend on the specific geometry of the uniform thin sheet and the axis specified. If more information about the shape and the axis is provided, a more accurate formula can be given. The calculation could yield $I = \frac{1}{12} M (b^2 + h^2)$ for rectangles, or similar formulas for other shapes.