Centroids Summary
The Centroid (or Centre of Gravity) of a shape is the point where it balances – the average location of its weight. Imagine trying to balance any shape on the tip of your finger, the point where your finger would need to be for it to balance, is the centroid. If a shape has uniform weight then the centroid will be in the centre.
To find the position of the centroid find: average position of weight along the x-axis (x)
and: average position of weight along the y-axis (y)
We find these averages by taking moments: about the y-axis to find x
about the x-axis to find y
Remember: moments = force x perpendicular distance
To find the centroid we do:
Total force x average distance = sum of individual forces x their distance
e.g. Find the centroid of a system of 3 particles: 6g located at (1,1), 4g located at (3,3) and 1g located at (4,2). (This question could also be about a lamina with weights placed in these positions – the method is exactly the same.)
1st do a sketch:
Total force x average distance = individual forces x distances
Equate moments about the y-axis: 11g×x=6g×1+4g×3+(1g×4)
(i.e use distance from y-axis) 11gx=22g
x=2
Equate moments about the x-axis: 11g×y=6g×1+4g×3+(1g×2)
(i.e use distance from x-axis) 11gy=20g
y=1.8
Using Calculus:
If the lamina is bound by a curve then you would use calculus to find the centroid. E.g find the centroid of the region bounded by y=f(x) and the lines x=a and x=b.
x=xfxdxfxdx y= fx22dxfxdx
Notice that there are 3 integrations with limits to calculate. The denominator is the same for each so you only need to do that once. In order to find the centroid you need to calculate:
fxdx (integrate the function with the limits)
xfxdx (multiply the function by x and integrate with limits)
f(x)22dx (square the function and integrate with limits.
You could put the 12 outside the integral for ease)
For example:
Find the centroid of the region bounded by y=x2 and the lines x=1 and x=3.
(Replace every f(x) with x2 and do the 3 integrations with limits)
x2dx = 263 (or 8.7)
xx2dx = x3dx =20
x222dx =12 x4dx = 1215 (or 24.2)
Now we can substitute into the formulae to find that:
x=208.7=2.3
y=24.28.7=2.8
H Jackson 2013 / Academic Skills 1