ABSTRACT
9.1.1
The moment of inertia of a ring is the mass times radius squared. I water = M r 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493003/15deefc7-84a6-4935-a96b-0bf4a637cb76/content/eq879.tif"/>
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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493003/15deefc7-84a6-4935-a96b-0bf4a637cb76/content/eq880.tif"/>
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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493003/15deefc7-84a6-4935-a96b-0bf4a637cb76/content/eq881.tif"/>
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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493003/15deefc7-84a6-4935-a96b-0bf4a637cb76/content/eq882.tif"/>
