how can ı find the moment of inertia in the mecanic system?

Answer 1

Your question is not clear to me. Do you mean derive the MOI mathematically or calculate the MOI experimentally?

If you mean experimentally, hang your system by its center of mass (which can also be found experimentally) and apply torque = Fr to it at multiple places and record the angular acceleration alpha each time. F is the force and r is the distance to the CM where the system axis of rotation is for each test.

MOI = Fr/alpha; where torque Fr is applied and alpha is measured; so you can find MOI. Do this around the x, y, and z axes of your system. This will give you MOI in the three dimensions and consequent three axes of rotation. And when specifying MOI, the axis of rotation must be specified because MOI changes according to the orientatioin of the rotation if the body is not perfectly symmetric around all three dimensions (e.g., like a sphere is).

If you mean mathematically, consider I alpha = Fr; where I is the MOI. For a point mass m, we have I alpha = mar; where F = ma is well known. Thus I = mar/alpha but a = alpha r; so that I = m alpha r^2/alpha = mr^2 is the moment of inertia for a point mass m revolving around an axis of rotation r distance away.

We can build on the point mass MOI by noting all solid objects, including your system, are nothing more than an accumulation (or sum) of point masses. Thus, the MOI of a system (around a specified axis of rotation) is just the integrated sum of all the point mass MOI's.

How that integration would be done depends on the shape, density, and such of the body. So I can't give you a specific solution. But if you go to http://en.wikipedia.org/wiki/List_of_moments_of_inertia, you'll find a nice long list of inertias for some more or less regular shaped systems.

Answer 2

If the system is made of point masses the total MOI is the sum of the individual MOI of each mass.

For extended bodies you have to integrate. Look at the pictures below and see that there are formulas for certain geometrical shapes.

For more about rotational dynamics and MOI see the second source

Answer 3

It depends on what the system consists of. The article below might be of help.