A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

See explanation

Explanation:

Solution:-

- A rectangular sheet of side length ( a and b ) with mass ( M ) has a moment of inertia about a perpendicular axis pass through its centroid ( Ic ) given as follows:

- We are to determine the moment of inertia of the rectangular sheet of metal about an axis perpendicular to the plane passing through either one of the corners.

- For this we will apply the parallel axis theorem. The theorem translates the axis of moment of rotation by a distance ( d ) from its centroid and allow us to determine the moment of inertia of any object about arbitrary point in space provided the moment of inertia of the object is known at its centroid ( Ic ).

- The parallel axis theorem express this with the following relation:

- First we will determine the distance ( d ) between the center of mass and any corner of the metal sheet. We will assume that the material is of uniform density. Hence, center of mass = geometric center of an rectangle.

- We will use Pythagorean theorem to determine the distance ( d ) - diagonal length as follows:

- Use the given moment of inertia about its centroid and the distance ( d ) calculated above with the help of parallel axis theorem we have: