Moment of Inertia of Half Ring (Half Circle)

[Arman777]

Homework Statement

there's an object which makes a pendulum motion.Lets suppose we hang the mass to the ceiling.We released the object with inital angle 0 to the ceiling.(I mean the angle between the object and the ceiling is zero).Whats the moment of the Inertia to the point A.

A is a point on the ceiling which middle of the motion.Lets suppose Lenght of the rope is L then.Imagine the inital position.there's mass m and it makes zero angle with object.Now A point will be (Objects position-L)=A

Mass of object m
Lenght of rope L

Homework Equations

I=∫mr^2dr

The Attempt at a Solution

M=∫mdr which r goes to -r to r
and
I=∫mr^2dr so
I=∫dM/dr r^2 dr
I=∫dMr^2
I=Mr^2

Is this correct.

 

[BvU]

Yes. All the mass is at distance r from the rotation axis. You basically assume your object is a point mass.

If this assumption no longer holds, the parallel axis theorem is a good tool to use.

 

[haruspex]

Your algebra is sort of correct if you define m to be the linear density of the object (as a function of distance along it) and M as the total mass. Even then, you have used M in two different ways. Inside the integral you have used it as total mass from the axis out to some distance r. Outside the integral you used it as the mass of the complete object.