Suppose a rod has a density of p(x)=x+3, where x is the distance from the left end of the rod. If the center of mass is at x=3, determine the length and mass of the object.

I only know how to solve these if I'm given density plus mass or length I'd love for someone to help me start this since I'm given center of mass instead. I could solve mass if I knew the bounds of the integral but without length that's impossible.

Possibly a tip in it for who helps me:)




e/ be back in an hour to see results. Thanks for any help. xoxo

This post was edited by Elementis on Oct 10 2013 09:52am

well the rod starts at x=0 so
the bounds of the "left integral" will be from 0 to 3
the bounds for the "right integral" will be 3 to a, where a is the length of the rod

This post was edited by Azrad on Oct 10 2013 10:00am

To find the centroid of any rod, or in this case the center of mass (x = 3), you can use the equation as follows...I'll try to type this out as best I can.

Center of mass = Integral(density*location)/Integral(density) where density is some weight per unit length

or basically

centerofmass = integral(rho*s*ds)/integral(rho*ds)

3 = integral[(x+3)(x)dx]/integral[(x+3)dx]

3 = [x^3/3 + 3x^2/2]/[x^2/2 + 3x] where x is going to be your length since the integral starts at zero.

Solve for x.



That also works.

Just as a sanity check, since you have an increasing density with the length of the rod, your length should be somewhere in the neighborhood of 4 or 5, but not 6.

This post was edited by Dontrunaway on Oct 10 2013 10:01am

I think I like that way better^^^.

but for completeness these are the integrals I was discussing:


This post was edited by Azrad on Oct 10 2013 10:19am