How to solve

$(25$ pts$)$ A hydrometer is a simple device used to measure the density of a liquid $($you

may have used something similar to measure the density of the antifreeze coolant in your

car engine$).$ A simple version consists of a test tube of outer diameter $D$ with a weight on

the bottom. First it is placed into pure water of known density ${}_w,$ and it sinks part way. A

red mark is drawn on the tube at the location of the water surface, height $h_w$ from the

bottom of the tube. Next it is placed into a liquid of unknown density ${}_u,$ and it sinks to a

shorter distance, as sketched, height $h_u$ from the bottom of the tube; a blue mark is drawn.

$($a$)$ Without doing any calculations, is the density of the unknown liquid less than, equal

to$,$ or greater than that of water? Explain.

$($b$)$ Derive an expression for unknown density ${}_u$ as a function of the other variables, as

needed. Stay in variables and show all your work.

$($c$)$ Plug in the following numerical values and calculate the density ${}_u$ of the unknown

liquid: $D=1\mathrm{.}45cm,{}_w=1000k\frac{g}{m^3},h_u=9\mathrm{.}57cm,$ and $h_w=10\mathrm{.}38cm.$