In Example 12.2, suppose you wanted to calculate the center of mass of the rod using Equation 12.5. What coordinate system must you use for your calculation?

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We start by considering what Equation 12.5 requires: a coordinate x that measures positions along the rod from a chosen origin, so the mass distribution can be integrated to find the center of mass.
Option 1: 'A coordinate system that has an origin located at the left end of the rod.' This is the natural and most convenient choice because it makes x measure distance from the left end, aligning with the common setup for a uniform rod and simplifying the integral for x dm. It yields x_cm = L/2 for a uniform rod, if the mass distribution is symmetric, and is consistent with using Equation 12.5.
Option 2: 'This calculation is independent of the coordinate system you choose.' While the physical center of mass is an invariant quantity, the form of the integral and the numerical value of x_cm depend on how x is defined relative to the chosen origin. Saying the calculation is independent of the coordinate system is misleading, because you must specify an origin to define x and perform the integral.
Option 3: 'A coordinate system that has an origin located at the middle of the rod.' Choosing the middle as origin would still allow computation, but it changes the expression for x and requires adjusting the limits and the integrand in Equation 12.5. The problem asks which system must be used for the calculation, and the middle is not the standard or required origin for this equation.
Option 4: 'A coordinate system that has an origin located at the right end of the rod.' Similar to option 3, you could use the right end as origin, but it is not the conventional or necessary choice for applying Equation 12.5 to this rod. It would simply shift the x-values and still yield the same physical center of mass after translation, but it is not the specified required origin.
In summary, Option 1 provides the origin that aligns with the standard application of Equation 12.5 for a rod measured from its left end, making the calculation straightforward and unambiguous. The other options either misrepresent the requirements of the equation or introduce unnecessary shifts of origin that are not mandated by the problem.

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

In Example 12.2, suppose you wanted to calculate the center of mass of the rod using Equation 12.5. What coordinate system must you use for your calculation?

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