Moment of Inertia: Find Answers to 2 Problems

[mr_joshua]

There are 2 questions that are giving a few of us problems.

1. Find the moment of inertia of a rod about an axis through its center if the mass per unit length is A=A*x A is linear density. This is a simple problem, but we are not sure how to use the given value for A (linear density). so far we have I=A * intergral of x^2 dx from 0 to L/2. which you finally get 1/12ML^2...



2. Same type of inertia problem except this is a disk about its center where mass density Q (which in normal cases is dm/dA) here Q=Q*r^(-1)

Please help.

 

[e(ho0n3]

The statements
- A = A*x
- Q = Q*r^(-1)
don't make any sense.

Remember the definition of moment of inertia:

\int_V \rho(\mathbf{r})dV(r\sin\theta)^2

where θ is the angle between r and the axis of rotation and ρ(r) dV is the mass element at location r. If ρ(r) isn't constant you can't take it out of the integral.

 

[M98Ranger]

For the first problem, the moment of inertia of a rod about an axis through its center can be found by using the formula I = A * integral of x^2 dx from 0 to L/2, where A is the linear density and L is the length of the rod. This formula assumes that the rod has a uniform density along its length. To use the given value for A, you simply substitute it into the formula instead of using a variable. So, the final answer would be I = A * (L/2)^3 / 3, which simplifies to 1/12 * A * L^2. This is the same answer you got, but it's important to note that A is not a variable, but a given value.

For the second problem, the moment of inertia of a disk about its center can be found by using the formula I = Q * integral of r^2 dA, where Q is the mass density and r is the distance from the axis to the element of area dA. In this case, Q is given as Q = Q * r^-1, which means that Q is inversely proportional to r. This can be rewritten as Q = k/r, where k is a constant. Substituting this into the formula, we get I = k * integral of r dA from 0 to R, where R is the radius of the disk. This integral can be simplified using the formula for the moment of inertia of a disk, which is I = 1/2 * m * R^2, where m is the mass of the disk. So, the final answer would be I = k * 1/2 * m * R^2, which simplifies to 1/2 * Q * m * R. Again, it's important to note that Q is not a variable, but a given value.