How do I determine the center of mass for a rod with varying linear density?

[aborder]

Homework Statement

A rod of length 36.00 cm has linear density (mass per length) given by
λ = 50.0 + 21.5x

where x is the distance from one end, and λ is measured in grams/meter.

A. Find Total Mass
B. Find center of mass from x=0

Homework Equations

1/M(integral)xdm , where M is total mass

The Attempt at a Solution

So I figured the total mass was the just the integral of the given density equation. Which was correct and part A was found to be 50x + 21.5x^2/2. Substituting in makes gets 19.39g (the right answer.) But I am confusing how to find the total mass with the necessary calculus requred to determine the center of mass. The issue I am having is with determining the center of mass.

 

[NascentOxygen]

Can you find an x-value where the mass to its left = the mass to its right?

 

[aborder]

Can you find an x-value where the mass to its left = the mass to its right?

That sounds like a good idea conceptually, but I'm not sure how to figure this mathematically.

 

[NascentOxygen]

Integral from 0 to x0 = integral from x0 to 0.36

 

[aborder]

Integral from 0 to x0 = integral from x0 to 0.36

Well that worked most excellently. Thank you. How did you see to do that?

 

[NascentOxygen]

I guess I've encountered a problem like that once or twice before.

I found calculus to be a breath of fresh air. :!) :!) The things it can do seemed limitless. Calculus is like the Swiss army knife of mathematics--a tool with no limit to its practical uses, as well as satisfying endless hours of intellectual amusement.

Unfortunately, I've forgotten most of the techniques, and now just retain admiration for the concept.

 

[GrantB]

Well that worked most excellently. Thank you. How did you see to do that?

There is a point on the rod where the mass to the left equals the mass on the right. He chose an arbitrary point where this is the case. Setting the integrals equal to each other and solving like you did for total mass, then using Algebra you should be able to solve for x0.