Moment of Inertia for a Disk with Hole

AI Thread Summary
To find the moment of inertia of a circular disk with a hole, first calculate the moment of inertia for the entire disk using I = 1/2 MR^2. Then, determine the mass of the hole based on its area relative to the disk's area. Apply the parallel axis theorem to account for the hole's position by subtracting its moment of inertia from the disk's moment of inertia. The final calculation confirms the correct approach, ensuring the moment of inertia is evaluated relative to the origin of the disk. This method leads to an accurate solution for the modified disk's moment of inertia.
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Homework Statement



A uniform circular disk has radius 36 cm and mass 350 g and its center is at the origin. Then a circular hole of radius 7.2 cm is cut out of it. The center of the hole is a distance 10.8 cm from the center of the disk. Find the moment of inertia of the modified disk about the origin.

Homework Equations



I=.25mr^2 (I think...!)

The Attempt at a Solution



I tried finding the moment of inertia for both the hole and the disk and subtract the hole, but that didn't work.
I'm not sure if that equation I posted is correct, I just googled it.
How am I supposed to find the mass for the moment of inertia equation?
At this point I just wish there was a straight up equation for it! Please help!
 
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I = 1/2 MR2

You need to subtract the inertia of the hole about the axis through the center of the disk.

(Parallel Axis theorem: I = Ic +md2)
 
How would I find the mass of the hole in that case?

If I use I=.5MR^2 for the entire disk (not counting the hole), I get I = 226800 g cm^2.

EDIT: Okay I figured out the mass of the hole. For the parallel axis theorem, would Ic be the moment of inertia for the whole disk? I'm just not sure how to apply it. Thanks again!
 
Last edited:
Wait, never mind I got it!

Once I got the mass by using the proper ratio: (350/[Area of disk - Area of hole]) = Mhole/[Area of hole], I just did the following:

I (of disk without hole) - .5(Mhole)(Rhole)^2
and got my answer!

Thanks!
 
Are you sure it is the correct answer? Remember you need to consider the moment of inertia of the hole relative to the origin of the disc.
 
It's correct, it's an online homework program that tells you whether it is or not =)
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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