course Phy 231
I had a question about this problem:********************
A uniform disk is growing in such a way that its radius increases by .7 cm every minute. The mass density of the disk is 5 grams per cm^2. If I(r) is the moment of inertia of the disk when its radius is r, then what are dI / dr and dI / dt at the instant the radius is 35 cm?
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So I wasn't really sure how to do this one, but it seems like since
I = mr^2 and the area = pi(r)^2, then the mass would be .005(pi)(r)^2 since the density is 5g/cm^2.
So I = mr^2 = 5pi(r)^2 = 5pi(r)^4
So then the derivative would be dI/dr = 20pi(r)^3
and @ r = 35cm, it'd be a change in inertia of about 2693916 (g*cm^2) per cm change in radius.
then since it changes .7cm/min, .7 cm change in radius *(263916 g*cm^2) per cm change in radius) =~ 1885741 g*cm^2 change in inertia per minute
or 31429 g*cm^2 change in inertia per second.
Is that how you do this problem? I could have totally made that up, but I wasn't really sure. "
So I wasn't really sure how to do this one, but it seems like since
I = mr^2 and the area = pi(r)^2, then the mass would be .005(pi)(r)^2 since the density is 5g/cm^2.
So I = mr^2 = 5pi(r)^2 = 5pi(r)^4
So then the derivative would be dI/dr = 20pi(r)^3
and @ r = 35cm, it'd be a change in inertia of about 2693916 (g*cm^2) per cm change in radius.
then since it changes .7cm/min, .7 cm change in radius *(263916 g*cm^2) per cm change in radius) =~ 1885741 g*cm^2 change in inertia per minute
or 31429 g*cm^2 change in inertia per second.
Is that how you do this problem? I could have totally made that up, but I wasn't really sure.
Very good.
Just to check the symbolic calculation would be
mass = rho * A = rho * pi r^2, where rho stands for density.
Thus
I = m r^2 = rho * pi r^4
and
dI/dr = 4 pi rho r^3.
By the chain rule, dI/dt = dI/dr * dr/dt = 4 pi rho r^3 * dr/dt.
dr/dt is .7 cm/min, r = 35 cm and rho = 5 g/cm^2. Substituting we get the same result you obtained, in g cm^2 / minute.
Converting to g cm^2 / s is very easy, and your result holds up completely.
Nice work.