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Grade upto college level Mechanics

Fig. shows a symmetrical rigid body rotating about a fixed axis. The origin of coordinates is fixed for convenience at the center of mass. Prove, by summing over the contributions made to the angular momentum by all the mass elements mi into which the body is divided, that src=data:image/png;base64,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 is the total angular momentum.
src=data:image/png;base64,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

Profile image of Amit Saxena
11 Years agoGrade upto college level
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1 Answer

Profile image of Navjyot Kalra
11 Years ago
The figure below shows the symmetrical rigid body rotating about the fixed axis:
232-1117_1.PNG
The angular momentum of the rigid body can be given in terms of its mass element as:
232-655_1.PNG
232-953_1.PNG
232-487_1.PNG
232-1394_1.PNG