Find a mass density function so that , the center of mass of a rod of length , is at a distance from one end.

Let

Then we compute the center of mass of the rod,

Thus, the center of mass is a distance from one end.

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Find a mass density function so the center of mass is at L/4

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Find a mass density function so that , the center of mass of a rod of length , is at a distance from one end.

Let

Then we compute the center of mass of the rod,

Thus, the center of mass is a distance from one end.

x^(-2/3) also works, doesn’t it? for the other side

How did you know that the density should be x squared.. I mean is there a method for that or you have guessed it?

L/4 from the end which mean 3L/4 from the start. then. 3L/4 = 3x^4/4x^3 from 0 to L=> int(x^3)/int(x^2)=int(x*p(x))/int(p(x)). =>p(x)=x^2.

hope it helps

Probably the easiest way to guess that the density function should be x^2 is to look back to Exercise 20. In that exercise, the density function was given as x^2, and you calculate the center of mass as 3L/4, which is L/4 from one end of the rod. So, Apostol already gave you the solution, just two exercises earlier.