Moment of inertia for composite object

[vetgirl1990]

Homework Statement

Find the moment of inertia for this composite object:
M1=5kg at (3,4) and M2=3kg at (6,8)

Homework Equations

Moment if inertia for point particle = MR²

The Attempt at a Solution

I'm not sure if moment of inertia should be componentized -- broken up into x and y components.
Ix =9M1 + 36M2
Iy= 16M1+64M2
Plugged in known values for m1 and m2

Now, should the components of moment if inertia now be summed (as you would if finding the total moment of inertia of say, a person sitting on a rotating disk) or I(total) = sqrt(Ix^2 + Iy^2), since they are lying in an axis?

 

[jbriggs444]

The moment of inertia is not broken into components. At least not in the context of rotation in a plane about an axis perpendicular to that plane. In introductory physics courses you will be dealing with rotation in a plane and can treat the moment of inertia as a scalar. [More generally, it is part of a tensor]

One way of thinking about this is to look more carefully at the formula MR². The R there is a vector. It has x and y components which are offsets from the axis of rotation (or coordinate system origin). Squaring R is done by computing the vector dot product of R with itself. The dot product of two vectors can be computed by multiplying each of their respective components together and then adding up the results. So (3,4) ⋅ (3,4) = 9 + 16 = 25. The key point is that the dot product of two vectors is always a scalar.

 

[vetgirl1990]

The moment of inertia is not broken into components. At least not in the context of rotation in a plane about an axis perpendicular to that plane. In introductory physics courses you will be dealing with rotation in a plane and can treat the moment of inertia as a scalar. [More generally, it is part of a tensor]

One way of thinking about this is to look more carefully at the formula MR². The R there is a vector. It has x and y components which are offsets from the axis of rotation (or coordinate system origin). Squaring R is done by computing the vector dot product of R with itself. The dot product of two vectors can be computed by multiplying each of their respective components together and then adding up the results. So (3,4) ⋅ (3,4) = 9 + 16 = 25. The key point is that the dot product of two vectors is always a scalar.

This is an excellent explanation, thank you! If the particles at each set of coordinates aren't massless, how would I take that into account?
Say one particle is 2kg at (3,4) and the other is 1kg at (3,4). Would I then find the dot product of the two vectors, then multiply by the total mass?
So 25(1+2)=50

 

[jbriggs444]

Say one particle is 2kg at (3,4) and the other is 1kg at (3,4). Would I then find the dot product of the two vectors, then multiply by the total mass?
So 25(1+2)=50

Did you transcribe those numbers properly? Both of your point masses are at the same place.

Edit: You might want to check your math as well. 25(1+2) is not usually equal to 50.

 

[vetgirl1990]

Did you transcribe those numbers properly? Both of your point masses are at the same place.

Edit: You might want to check your math as well. 25(1+2) is not usually equal to 50.

Sorry, I was typing that out on the go and being careless.
(3,4) m=1kg and (4,5) m=2kg
=(3,4)⋅(4,5) = 12+20 = 32 (1kg+2kg) = 96

 

[jbriggs444]

Angular momentum is an additive property. That is to say that if you have two or more objects rigidly connected to each other then the angular momentum of the whole assembly is the sum of the angular momenta of each part taken separately. [This holds as long as you are consistent and use the same axis of rotation throughout]

So start with that. What is the moment of inertia of the first mass taken by itself? What is the moment of inertia of the second mass taken by itself?