Integral physics problem?
A cylindrical space station of radius R and length L has a density that varies with distance from its center:
p(r) = A/r
a) Evaluate the mass dM of an infinitesimally thick cylindrical shell of radius r.
b) Calculate the total mass M of the entire space station.
c) Calculate the mass of the space station using infinitesimally thick cylindrical cross-sections, showing that you get the same answer no matter how it is "chopped up".
Having some trouble on this one, don't even know how to start it. Any help would be nice.
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My attempt at this:-
Density(p) = Mass (M)/ Volume (V)
p=M/V
M = p x V
dM = p(r) x dV
a)
For an infinitesimally cylinrical shell then
Its area will be its circumference (2 x pi x r) times the thickness of the shell dr = (2 x pi x r) dr
the volume, dV of this shell will be this area times its length, L
dV = (2 x pi x r x L) dr
dM = p(r) dV
dM = A/r x 2 x pi x r x L dr
dM = 2 x pi x A x L dr
b)
Integrating:-
M = Int [dM] = (2 x pi x A x L) Int [dr] between lomits of 0 and R
M = (2 x pi x A x L) x [ r] limits of R and 0
M = (2 x pi x A x L) x [ R -0]
M = 2 x pi x A x L x R