How Long Does It Take for a Water Droplet to Move Radially on a Turbine?

[GaryTravis]

Homework Statement

A water droplet condenses at r0=0.80 m. The turbine is rotating at a fixed speed of N RPM.

You can treat the water droplet as a point mass of m=0.002 kg. Starting from rest at its formation point, it starts moving outward on the radial vane as shown. Its mass stays constant during this motion (no new condensation nor evaporation).
Calculate the time (in seconds) it takes for the droplet to move from the point of formation (ro) to the radius r.
r0= 0.08m
r= 4.13m
w= 402rad/s

Homework Equations

V = w√(r^2-r0^2) = dr/dt

The Attempt at a Solution

∫dt = ∫(dr/w√(r^2-r0^2))

dt is from 0->t and dr is ro->r

I know that this is the equation for determining the time taken but I do now how to to integrate that function. I have been told it is t= 1/w*ln(r+√r^2-r0^2). But when i attempted to calculate the time it was always wrong.

 

[SteamKing]

Why don't you show your calculations?

 

[NascentOxygen]

∫dt = ∫(dr/w√(r^2-r0^2))

Is the square root term in the numerator or denominator? If the latter, then you need another set of parentheses to show it.