9.1.1

The moment of inertia of a ring is the mass times radius squared. I water = M r 2

The rate of change of water is the rate at which water flows in minus the rate at which water flows out. M ˙ = Q total − K M

First, we make a differential equation for I
_{water}
I
˙
water
=
M
˙
r
2
=
Q
total
r
2
−
K
M
r
2
=
Q
total
r
2
−
K
I

which has solution I water ( t ) = ( I water ( 0 ) − Q r 2 K ) e − K t + Q r 2 K lim t → ∞ I water ( t ) = Q r 2 K