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

In summary, the conversation discussed finding the total mass and center of mass of a rod with a given linear density equation. The total mass was found by taking the integral of the density equation, while the center of mass was determined by setting the integrals of the mass on the left and right sides equal to each other and solving for the point where they are equal. Calculus was praised for its versatility and usefulness in solving such problems.
  • #1
aborder
14
0

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.
 
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  • #2
Can you find an x-value where the mass to its left = the mass to its right?
 
  • #3
NascentOxygen said:
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.
 
  • #4
Integral from 0 to x0 = integral from x0 to 0.36
 
  • #5
NascentOxygen said:
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?
 
  • #6
I guess I've encountered a problem like that once or twice before. :smile:

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. :approve: :approve: :approve:

Unfortunately, I've forgotten most of the techniques, and now just retain admiration for the concept. :blushing:
 
  • #7
aborder said:
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.
 

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

1. What is the center of mass?

The center of mass is a point in an object or system where its mass can be considered to be concentrated. It is the average position of all the mass in the object.

2. Why is it important to determine the center of mass?

Determining the center of mass is important in many fields of science, including physics, engineering, and astronomy. It helps in understanding the overall motion and stability of objects, as well as predicting their behavior under certain conditions.

3. How is the center of mass calculated?

The center of mass can be calculated by dividing the sum of all the individual masses in an object by the total mass of the object, and then finding the weighted average of their positions in space.

4. Can the center of mass be outside of an object?

Yes, the center of mass can be outside of an object if the mass distribution is not uniform. For example, in a hollow object, the center of mass may be located outside of the physical boundaries of the object.

5. How does the center of mass affect an object's motion?

The center of mass is an important factor in determining an object's motion and stability. If an external force is applied to an object, it will accelerate in the direction of the force, but also rotate around its center of mass. The location of the center of mass also affects an object's ability to balance and maintain stability.

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