Why is the End Area of Shell 2πrdr in the Moment of Inertia Formula?

AI Thread Summary
The end area of a shell in the moment of inertia formula is represented as 2πrdr, where 2πr is the circumference of the shell and dr is its infinitesimal thickness. This differential area is crucial for calculating the total moment of inertia by integrating over the radius. The integration process sums the areas of all these thin circumferences to derive the total circular area. Understanding this concept is essential for those beginning to learn integration. The discussion highlights the importance of grasping differential forms in physics calculations.
PSOA
icyl2b.gif


Why is the end area of shell 2\pi rdr?

Thanks in advance. :rolleyes:
 
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PSOA said:
icyl2b.gif


Why is the end area of shell 2\pi rdr?

Thanks in advance. :rolleyes:

It's the area expressed in differential form. To acquire the correct formula, you need to calculate the integral with respect to r.

Intuitively, the differential form means this : to describe an infinitesimal small area, take the circumference 2 times pi times r and make it a bit thicker, expressed by the dr. Integrating over r means that you sum up the area of all this different, very thin, circumferences to get the total circular area.

marlon
 
Last edited:
Oh, I got that! Thank you marlon! I am giving my first steps over integration. And congratulations for the award!
 
PSOA said:
And congratulations for the award!
What award ?
I just lost my guru badge to somebody else, lol.

marlon
 
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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