4 Point masses form a body - Inertia & Rotational Kinetics

[LadyMario]

Homework Statement

Four point masses form a rigid body (they are connected by massless rigid rods) given the positions M1= 3kg (2m, 4m, 0m) M2= 2kg (1m, -4m, 0m) M3= 1kg (10m, 2m, 0m) M4= 5kg (-5m, 2m, 0m)
Find:
A) Moment of inertia of this system when it rotates about x axis
B) Moment of interia of this system when it rotates about y axis
C) Total rotational kinetic energy in the (A) case when ω= 4 rad/s
D) Total rotational kinetic energy in the (B) case when ω= 4 rad/s

Homework Equations

Rotational K = 1/2Iω²

The Attempt at a Solution

I'm really terrible at things with center of mass but I believe we'd have to somehow find the total systems Center of Mass in order to find it's moment(s) of Inertia. However I don't know how to do this with the different axis' (x & y). And as I know from the formula, I can't solve for Rotational Kinetic energy without them

Help?

 

[SteamKing]

Do you know how to calculate the center of mass of a body knowing the components of the body and the c.m. of each component? Do you know the formula for calculating center of mass? In cases such as this, try drawing a picture showing each component mass and its location from the origin.

You will have to deal with this before proceeding to calculate moment of inertia for the composite body.

 

[LadyMario]

Do you know how to calculate the center of mass of a body knowing the components of the body and the c.m. of each component? Do you know the formula for calculating center of mass? In cases such as this, try drawing a picture showing each component mass and its location from the origin.

You will have to deal with this before proceeding to calculate moment of inertia for the composite body.

I have the general idea, and I have drawn a diagram. I believe the formula is Xcm(M)=x1m1+x2m2+x3m3+x4m4 where M is the total mass and Xcm is the centre of mass in the x direction. But once I find this how do I get the moment of inertia for rotating around the x axis? Do I just use Xcm in the formula I=Icm + MD² (parallel axis theorem?) And if so what would be D because none of them fall right on the x axis...

 

[SteamKing]

D is going to be the distance of the c.o.m. of each point mass from the x-axis. (Hint: remember the Pythagorean Theorem). Calculate MOI about the origin for the body, then transfer the MOI from the origin to the c.o.m. using the parallel axis theorem.