Calculating Mss of a Rod using Integration

In summary, the conversation discussed finding the total mass, center of mass, and moment of inertia of a rod with a given density function. The solution involved using the integral of a polynomial and solving for the mass by integrating over the entire length of the rod. The center of mass and moment of inertia were also found by using appropriate equations.
  • #1
mm391
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Homework Statement



Let a rod with length L and constant cross-sectional area A have the density
ρ(x) = ρ((3.x^2 + 2.x.L)/L^2)

0 ≤ x ≤ L

where x is the distance from one end of the rod and ρ0 is a real constant.
(a) Find the total mass M of the rod.
(b) Find the x-coordinate xc of the centre of mass of the rod.
(c) Find, in terms of M, the moment of inertia of the rod about the vertical axis through x = 0.


Homework Equations



M = ∫Aρ(x).dx

The Attempt at a Solution



M = ∫Aρ(x).dx

M= A.ρ∫((3.x^2 + 2.x.L)/L^2).dx

I am slightly confused where to go from here! Do I use integration by parts?
 
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  • #2
Why? That is just the integral of a polynomial. What is the integral of [itex]3x^2 dx[/itex]? What is the integral of [itex]2x dx[/itex]?
 

1. How do you calculate the mass of a rod using integration?

To calculate the mass of a rod using integration, you need to integrate the density function of the rod over its length. This will give you the total mass of the rod.

2. What is the formula for calculating the mass of a rod using integration?

The formula for calculating the mass of a rod using integration is:
M = ∫ ρ(x) dx from x=a to x=b
Where M is the mass, ρ(x) is the density function, and a and b are the limits of integration (the length of the rod).

3. Can you explain the concept of integration in calculating the mass of a rod?

Integration is a mathematical technique used to find the area under a curve. In this case, the curve represents the density function of the rod. By integrating the density function over the length of the rod, we can find the total mass of the rod.

4. What are the units used in calculating the mass of a rod using integration?

The units used in calculating the mass of a rod using integration will depend on the units of the density function. If the density function is in kilograms per meter (kg/m), then the mass will be in kilograms (kg). It is important to make sure that all units are consistent when performing the calculation.

5. Are there any limitations or assumptions when using integration to calculate the mass of a rod?

One limitation is that this method assumes that the density of the rod is constant along its length. If the density varies, then the calculated mass will not be accurate. Additionally, this method may not work for complex shapes or objects with irregular density distributions.

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