The moment of inertia of a circular area about a tangent to the circle is calculated as the moment of inertia of the circular area about its centroidal axis in the plane of the lamina ______ (where r is the circle radius).

Explanation:

Moment of Inertia of a Circular Area About a Tangent to the Circle

The moment of inertia (I) of a circular area about a tangent to the circle is calculated using the parallel axis theorem. The theorem states that the moment of inertia of an area about any axis parallel to the centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the area and the square of the distance between the two axes.

Centroidal Moment of Inertia:

The moment of inertia of a circular area about its centroidal axis in the plane of the lamina is:

I_centroid = (π × r⁴) / 4

Parallel Axis Theorem:

Using the parallel axis theorem, the moment of inertia about a tangent to the circle is:

I_tangent = I_centroid + A × d²

Where:

• I_centroid = Moment of inertia about the centroidal axis
• A = Area of the circular lamina
• d = Distance between the centroidal axis and the tangent axis (equal to the radius r of the circle)

Calculation:

The area (A) of the circular lamina is given by:

A = π × r²

Substituting the values:

I_tangent = I_centroid + (π × r²) × r²

I_tangent = (π × r⁴) / 4 + π × r⁴

I_tangent = π × r⁴ × (1/4 + 1)

I_tangent = π × r⁴ × (5/4)

I_tangent = (5π × r⁴) / 4