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Last Modified: 2:01pm 10 Aug 11
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Last Modified: 2:01pm 10 Aug 11
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In real life, many objects are extended: they contain matter distributed somewhat uniformly over large distances. They don't look much like little compact balls, so it's not so easy to use the discrete approximation to calculate their centers of mass.
Consider the shape outlined on this piece of graph paper:
You CAN find the center of mass in the same old way, if you wish: treat each little square of material as an individual bit of matter with mass 1 unit.
In each group of students, set all but one student calculating the center of mass in this manner. You can split the work up (one person does x, one person does y), or check each other's work.
While they are performing the calculations, have the remaining student in each group find the center of mass in another way. Cut one of the shapes out of the paper with a pair of scissors. Then pick two points on the object, somewhere near two different corners. Gently poke a hole in the paper at each point. Suspend the object by a string through one hole, letting the object and the free end of the string hang straight down. Draw a line from the suspension point running straight downwards across the object. Then suspend the object from the other hole and draw a second line, again straight downward. Where the two lines cross should be the center of mass.
Compare the two approaches: does the calculation yield the same location for center of mass as the suspension?
But for really complicated objects, there's no way around it: in order to find the center of mass, you have to integrate.
Now it's your turn to find the center of mass of an extended object.
A long, thin triangle of length L and base H is made of a uniformly dense material. Where is its center of mass? (Hint: the location of the center of mass is easy to find in one direction using symmetry ... but you must do the integral to find the center of mass in the other direction).
Adapted from Prof. Michael Richmond.
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Last Modified: 2:01pm 10 Aug 11
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