L&T, 19.12-22
Again consider curve from to , divided into strips of thickness . The area of the strip is given by . The total area is given by the sum,
If the strip is very thin then all of it is approximately at a distance from axis. If we now add up NOT but instead times , i.e., “weighted” by , we get the first moment of the area about the axis,
This is usually called , even though it is the first moment around the axis.
Example 6.5:
Find the first moment of area under from to about the axis.
Solution:
Example 6.6:
Find the first moment of the area under from to about the axis.
Solution:
Integrate by parts: , , , . Therefore , and thus ,
Now consider the same strip of thickness . On this strip goes from to . Divide strip into segments of length as shown in Fig. 6.8 . The area of such a segment is . The total area of strip is . In the limit that becomes infinitesimal we get
as before. Now instead of summing segments we can weight each of them by the value of to get
To find we have to add the contributions of all strips
This is the formula for the first moment of the area about the x axis (This integral is same as that for the volume of revolution except for the factor outside the integral rather than ).
Example 6.7:
Find for area under curve from to (same area as in example xxxx(1))
Solution:
Therefore
For any plane shape with area , the centroid is a point with coordinates given by , , where is first moment of area about the axis, and is first moment of area about the axis.
Example 6.8:
Find the centroid of the area under from to using the previous two examples.
Solution:
We know that and , and we just need to determine ,
Therefore
If we have thin plate with constant thickness then the centroid is the position of centre of mass (C of M). The C of M is the point at which all mass can be regarded as acting. Let mass per unit area be : This will be constant if the thickness is constant (and material is of uniform composition). The total mass where is area. Turning effect about axis of mass at would be . A strip of thickness , height has area . Mass would be . Total turning effect is , therefore , therefore .