How Do You Find the Center of Mass for a Non-Uniform Bar Using Calculus?

[DLH112]

Homework Statement

find the mass of an unevenly distributed bar with length 2 meters whos mass at a point is given by an equation 0.6 + x^2. Then find the center of mass.

Homework Equations

The Attempt at a Solution

I got the first part (finding the mass) correctly, but I can't conceptually figure out why what i did to find the center of mass doesn't work.

integrate to get 0.6x + x^3/3 and evaluate from 0 to 2, to find that the mass is 2 kg.
What I tried to do to find the center of mass is just set that equation equal to half the mass and solve for X, however this is not correct and I do not know why...

 

[collinsmark]

Homework Statement

find the mass of an unevenly distributed bar with length 2 meters whos mass at a point is given by an equation 0.6 + x^2. Then find the center of mass.

Homework Equations

The Attempt at a Solution

I got the first part (finding the mass) correctly, but I can't conceptually figure out why what i did to find the center of mass doesn't work.

integrate to get 0.6x + x^3/3 and evaluate from 0 to 2, to find that the mass is 2 kg.
What I tried to do to find the center of mass is just set that equation equal to half the mass and solve for X, however this is not correct and I do not know why...

The approach you used would be correct if you were trying to find the location where 1/2 the mass is to the left and the other other 1/2 is to the right.

But that's not what you're being asked to find.

Here, the term "center of mass" needs to take into account the leverage involved. Sometimes the term is also called the barycenter, center of gravity, weighted center or point of balance. You're looking for the point such that if you were to grab the object at that point with your fingers, it would balance.

I'm guessing that you might find a formula for the barycenter (center of mass) from your textbook. With that, you can simply plug and chug.

Or you could derive the formula yourself. If you define r as the distance from the barycenter x_c to some small mass dm, then you need to set up an equation such that the sum of all r dm on one side of x_c equals the sum of all r dm on the other side of x_c. Then solve for x_c.

Good luck!