A rod of length has non-uniform linear mass density given by , where and are constants and . The value of for the centre of mass of the rod is at :

Question

A rod of length  has non-uniform linear mass density given by , where  and  are constants and . The value of  for the centre of mass of the rod is at :

Canvas Classes · Accepted Answer

🧠 Non-Uniform Rod — Direct Integration With the Standard CM Formula

Read this. A rod of length  with linear mass density . The density increases from  at  to  at . Find the -coordinate of the CM.

For a non-uniform 1-D mass distribution, the CM formula is

The numerator is the first moment of mass about the origin. The denominator is the total mass. Both integrals are elementary because  is a polynomial in .

First concrete move: expand , then compute the two integrals term by term. Each int

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