- #1
rbwang1225
- 118
- 0
Homework Statement
A small cylindrical test-tube of inner radius R is initially filled with a liquid up to height ##h_0##. The tube is connected by a long rope of length ##L## (##L»h_0##) and swinging horizontally with a constant angular velocity ##ω##. There is a tiny round hole of radius ##r## (##r « R##) at the bottom of the test-tube and the liquid is spraying out from the hole. What's the remaining height of the liquid as a function of time?
Homework Equations
Continuity equation and circular motion equations.
The Attempt at a Solution
##v\pi r^2=wR\pi R^2##
##v=\frac{R^3ω}{r^2},##
where v is the speed of the liquid just outside the hole.
The small cube moving out the hole is ##dx\pi r^2## which is equal to the lose of the liquid in the tube ##dh \pi R^2.##
##v=\frac{dx}{dt}##
Then the remaining height of the liquid as a function of time is ##h(t)=h_0-∫dh=h_0-∫\frac{r^2}{R^2}dx=h_0-∫_0^t\frac{r^2}{R^2}\frac{R^3ω}{r^2}dt=h_0-Rωt##
I know I might miss something, could anyone point out for me.
Regards.
Last edited: