Question

moment of inertia of a rectangle about its base

Answer 1

Let us consider one rectangular section ABCD as displayed in following figure. Let us assume that one line is passing through the base of the rectangular section and let us consider this line as line CD and we will determine the area moment of inertia for the rectangular section about this line CD.



B = Width of the rectangular section ABCD

D = Depth of the rectangular section ABCD

ICD = Moment of inertia of the rectangular section about the CD line

Now we will determine the value or expression for the moment of inertia of the rectangular section about the line CD

Let us consider one rectangular elementary strip with thickness dY and at a distance Y from the line CD as displayed in above figure.


 
Let us determine first the area and moment of inertia of the rectangular elementary strip about the line CD


Area of rectangular elementary strip, dA = dY.B

Moment of inertia of the rectangular elementary strip about the line CD = dA.Y2

Moment of inertia of the rectangular elementary strip about the line CD = B Y2 dY


Now we will determine the moment of inertia of entire area of rectangular section about the line CD and it could be easily done by integrating the above equation between limit 0 to D.


Therefore, moment of inertia of entire area of rectangular section about the line CD will be as displayed here in following figure



 
Therefore, moment of inertia of the rectangular section about the line CD

ICD =BD3/3

Answer 2

Rotational inertia is important in almost all physics problems that involve mass in rotational motion. It is used to calculate angular momentum and allows us to explain (via conservation of angular momentum) how rotational motion changes when the distribution of mass changes.