What is the derivation of the moment of inertia of a solid sphere?

AI Thread Summary
The discussion focuses on understanding the derivation of the moment of inertia of a solid sphere, specifically the origin of the (1/2)y² term. The user is confused about this term, associating y² with the radius but questioning the necessity of the 1/2 factor. Additionally, there is a query regarding different approaches to defining mass elements in the context of calculating moment of inertia, comparing methods used by a website and a physics professor. The conversation emphasizes the importance of clearly defining relationships in the derivation process to alleviate confusion. Understanding these foundational concepts is crucial for accurately calculating the moment of inertia.
piquante
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So here I just need help understanding a concept.
I was reading this website's derivation of the moment of inertia of a solid sphere here: http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph3
For the life of me I can't see where the (1/2)y2 term comes from. I would understand y2 because that would be the radius, but the 1/2?
I understand the other steps but I'm very much stuck on this first one.

A related question for anyone who has an opinion: this website starts out these problems by writing dI = abc dm. My physics professor tends to start with dm = dfg dr. Is there a best way to approach these problems? I'm always the most confused at the beginning when I need to define the above relationships. Thanks!
 
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To find the moment of inertia of the sphere, they had to sum the moment of inertia of many tiny disks of mass dm. The moment of inertia of a disk of mass M is (1/2)MR^2. To see a derivation of the moment of inertia of a disk: http://mikebloxham.com/H7A/I for sphere.pdf
 
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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