Centre of mass of a composite body
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Centre of Mass of a Composite Body
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A composite body is one made up of separate parts each of identifiable shape and mass and each having its own centre of mass. The aim is to find the centre of mass of the composite body. This is done by means of taking moments according to the principle that the moment of the whole body about its centre of mass is equal to the sum of the moments of each parts. Let K be a composite body comprising n parts of mass Mi, 0 = i = n. Let [Equation goes here - download the original to see it.] be the total mass of K. Let P be any turning point or L be any axis. Let xi be the distance of the centre of mass mi from K or the perpendicular distance of the centre of mass mi from L. Let be the perpendicular distance of M from P or L of the centre of mass of K. Then [Equation goes here - download the original to see it.] We proceed to apply this result to a number of examples. Example (1) A lamina is made up in the shape of the symbol as shown: [Diagram goes here - download the original to see it.] Find its centre of mass. Let [Equation goes here - download the original to see it.] be the centre of mass. The coordinates of the centre of mass of each piece are as indicated. [Diagram goes here - download the original to see it.] Labelling the segments as shown, their masses are mA = 5 mB = 3 mC = 2 units of mass. Total mass is M = mA + mB + mC = 5 + 3 + 2 = 10. Then, taking moments about Oy [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Taking moments about Ox [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] You can be asked to find the centre of mass created by the removal of one solid shape from another. The following example illustrates this type of problem and how it is solved. Example (2) A wooden bowl is made by removing a hemispherical portion of radius 3a from solid hemisphere of wood of radius 4a. [Diagram goes here - download the original to see it.] Given that the centre of mass of a uniform solid hemisphere of radius r is 3r/8 from its plane face, find the centre of mass of the bowl from the origin, O, along the axis ON. When the bowl is suspended from its rim its diameter makes an angle a with ON. Find a. Let x-bar be the position of the centre of mass of the bowl from O along the axis ON. The volume of a hemisphere of radius r is [Equation goes here - download the original to see it.] Hence the volume of solid hemisphere of radius 4a is [Equation goes here - download the original to see it.] And the volume of the cavity is [Equation goes here - download the original to see it.]Let ? be the density of the wood per unit volume. Then the mass of the solid hemisphere is [Equation goes here - download the original to see it.] The missing mass of the cavity is [Equation goes here - download the original to see it.] The mass of the bowl is [Equation goes here - download the original to see it.] The solid hemisphere can be viewed as made of two parts: the bowl and the smaller solid hemisphere removed from it. [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Centre of mass of a composite body
1 Centre of Mass of a Composite Body
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