The Chi squared distribution
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Centre of Mass of two-dimensional objects
Equations are omitted for technical reasons - download the original pdf
The centre of mass of an object is found using the principle of moments. About the centre of mass [Equation goes here - download the original to see it.] Example Find the centre of mass of an object comprising of two particles of mass 2kg and 5kg respectively, joined by a rod of no mass of length 1m. [Diagram goes here - download the original to see it.] We fix an origin at, say, the 2kg end and let x indicate the distance from this origin of the centre of mass. [Diagram goes here - download the original to see it.] The weights are 2g and 5g respectively. About the centre of mass the sum of the moments is zero. That is, Clockwise moment = Anticlockwise moment [Equation goes here - download the original to see it.] This principle, that about the centre of gravity the sum of the clockwise moments equal the sum of the anticlockwise moments leads to shortcut methods of finding centres of mass for objects that possess some obvious symmetry. We are here concerned with 2 dimensional objects. Consider a rectangular lamina of uniform density. Then clearly, for any line passing through the centre of mass, the mass on one side of the line must equal the mass on the other. [Diagram goes here - download the original to see it.] Mass of left half = Mass of right half For if the masses did not balance, the torque produced by one half would be greater than that produced by the other and the object would be rotating. The centre of mass of a uniform body must lie at the intersection of any two axes of symmetry. [Diagram goes here - download the original to see it.] This symmetry property is used to find the centre of mass of objects that have some obvious symmetry! The following table illustrates this principle, and gives the standard positions of centres of mass for simple objects. [Table goes here - download the original to see it.]
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Contents of
The Chi squared distribution
1 The chi squared Distribution
2 The Concept of a Centre of Mass
3 Centre of Mass of two-dimensional objects
4 Triangular lamina
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