Find the Mass Moment of Inertia about the x,y,z axes

[Northbysouth]

Homework Statement

The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x, y, z axes.

I have attached an image of the question

Homework Equations

The Attempt at a Solution

The answer is given but I don't understand how the answer is reached.

For I_xx The slender bar of length 2L has no mass moment of inertia about x. Then the other two rods each have a mass moment of inertia of 1/3*ml² or do these cancel out and it's only the spheres which give the system a mass moment of inertia?

I think that, for the spheres, the mass moment of inertia should be:

2*(2/3)mr² + 2ml²

Is the first portion, 2*(2/3)mr², 0 because the radius is negligible small?

This would leave me with 2mL²

Any guidance would be appreciated. Thank you.

 

[SammyS]

Homework Statement

The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x, y, z axes.

I have attached an image of the question

Homework Equations

The Attempt at a Solution

The answer is given but I don't understand how the answer is reached.

For I_xx The slender bar of length 2L has no mass moment of inertia about x. Then the other two rods each have a mass moment of inertia of 1/3*ml² or do these cancel out and it's only the spheres which give the system a mass moment of inertia?

I think that, for the spheres, the mass moment of inertia should be:

2*(2/3)mr² + 2ml²

Is the first portion, 2*(2/3)mr², 0 because the radius is negligible small?

This would leave me with 2mL²

Any guidance would be appreciated. Thank you.

Anyone helping you will want a nice size image of that.

How far is each mass from each of the three possible axes of rotation?

.

 

[Northbysouth]

So, I'm only really using the parallel axis theorem?

For the mass moment of inertia about the x axis, the centers of the slender rods perpendicular to the x-axis are a distance L/2 away and the spheres are each a distance L away.

But summing these would give me 3mL^2

 

[SammyS]

So, I'm only really using the parallel axis theorem?

For the mass moment of inertia about the x axis, the centers of the slender rods perpendicular to the x-axis are a distance L/2 away and the spheres are each a distance L away.

But summing these would give me 3mL^2

Light rods usually implies that you can ignore the mass of the rods. After all, that mass is not given.

 

[Northbysouth]

Ahh, that makes sense now. I think I misread the question, because I was thinking that the slender rods also had mass m.

But, then isn't it the same case for I_yy? The spheres are still a distance L from the y axis.

 

[SammyS]

Ahh, that makes sense now. I think I misread the question, because I was thinking that the slender rods also had mass m.

But, then isn't it the same case for I_yy? The spheres are still a distance L from the y axis.

Not for the y-axis, look again.

What are the coordinates of the two masses?

 

[Northbysouth]

On the x-z plane, the spheres have coordinates (L, L) and (-L, -L). Should I use Pythagoras here? Or am I using mL^2 twice for each sphere to account for the x and z components?

 

[SammyS]

On the x-z plane, the spheres have coordinates (L, L) and (-L, -L). Should I use Pythagoras here? Or am I using mL^2 twice for each sphere to account for the x and z components?

Use Pythagoras.

I would write the coordinates as ordered triples, i.e. (L, 0, L) and (-L, 0, -L) .