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
66
0

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?
 
Physics news on Phys.org
  • #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]?
 

FAQ: Calculating Mss of a Rod using Integration

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.

Similar threads

How can I calculate the cumulative mass of a disk using disk mass density?
Replies
25
Views
2K
Force due to two thin charged rods acting on each other
Replies
7
Views
2K
Verify Green's Theorem in the given problem
Replies
10
Views
1K
Prove limit comparison test for Integrals
Replies
2
Views
1K
Improper integral of a normal function
Replies
7
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
I A rod that falls while rotating from the end of a table
Replies
2
Views
462
Line Integral of circle in counterclockwise direction
Replies
5
Views
1K
Thin Rod Gravitational Potential and Field Vector
Replies
6
Views
2K
Centroid calculation using integrals
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top