In Example 12.1, if a third ball was placed on the rod a x=10cm, how would the center of mass of the system change?

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The prompt asks us to assess how the center of mass (COM) of the system would change when a third ball is placed on the rod at x = 10 cm.
Option 1: 'The center of mass position would stay the same' — This would be true only if the third ball had zero mass or if its mass and position contributed no net shift to the overall COM. In general, adding any finite-mass object at a finite position will shift the COM unless a perfectly compensating arrangement exists. Therefore this statement is not reliably correct without specific mass information.
Option 2: 'The answer depends on the mass of the third ball' — This is a precise and correct qualifier. The COM of a system is the mass-weighted average of all positions. Introducing a third mass at x = 10 cm will shift the COM by an amount that depends on the third ball’s mass relative to the masses of the existing components and their positions. If the third ball is very small, the shift is small; if it’s large, the shift is larger, either toward or away from the existing masses depending on their configuration. This captures the essential dependency on mass.
Option 3: 'The center of mass position would move to the right' — Without knowing the relative placements and masses of the existing components, we cannot assert a definite direction of the shift. The COM could move left or right (or not shift much) depending on the mass and exact geometry. Therefore this blanket directional claim is unfounded.
Option 4: 'The center of mass position would move to the left' — For the same reason as Option 3, asserting a fixed leftward shift is not generally correct without information about masses and positions. This makes the option too specific and likely incorrect in the general case.
In summary, the only option that accurately reflects the fundamental dependence on the third ball’s mass (and its interaction with the existing system) is Option 2. The other statements make unconditional claims about the shift or direction that require additional mass/position details to be valid.

PHYS 2305 Spring 25 Lin Quiz 09 - Chapter 12 (12.1-12.4,12.9)

In Example 12.1, if a third ball was placed on the rod a x=10cm, how would the center of mass of the system change?

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