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Physics / Classical Mechanics 0 / Chapter 3 / Problem 3.37

Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor

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A system consists of N masses ma at positions ra relative

Chapter 3, Problem 3.37

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QUESTION:

A system consists of N masses ma at positions ra relative to a fixed origin 0. Let ra denote the position of ma relative to the CM; that is, rat = ra R. (a) Make a sketch to illustrate this last equation. (b) Prove the useful relation that > mares = 0. Can you explain why this relation is nearly obvious? (c) Use this relation to prove the result (3.28) that the rate of change of the angular momentum about the CM is equal to the total external torque about the CM. (This result is surprising since the CM may be accelerating, so that it is not necessarily a fixed point in any inertial frame.)

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QUESTION:

A system consists of N masses ma at positions ra relative to a fixed origin 0. Let ra denote the position of ma relative to the CM; that is, rat = ra R. (a) Make a sketch to illustrate this last equation. (b) Prove the useful relation that > mares = 0. Can you explain why this relation is nearly obvious? (c) Use this relation to prove the result (3.28) that the rate of change of the angular momentum about the CM is equal to the total external torque about the CM. (This result is surprising since the CM may be accelerating, so that it is not necessarily a fixed point in any inertial frame.)

ANSWER:

Step 1 of 5

Consider a system of number of masses each of mass and the position of each mass with respective to the origin is . If the position of a particle with respective to the center of mass is , and the position of center of mass with respective to the origin is .

Draw the figure for the above situation.

In the above figure, the point represents the point of any mass in the system, and represents the point of center of mass.

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