a horizontal cylindrical boiler 6 ft. in diameter is half full of water.

Question

find the force on one end.

Answer 1

Hi Darren D.,

A great problem for a straightforward conversion to an integral!

The pressure vs. depth equation is: P = hρg . We may neglect the contribution of the atmosphere to the absolute pressure, since it is additive to that generated by the fluid head and balanced by air pressure external to the tank (unless in a vacuum surround, then add 1 atm*(area of tank end)).

So, integrate from 0->3 ft depth (let's call this the x direction), the pressure times the linear d(depth) slice length, obtained from the circular rim equation: x^2 + y^2 = 3^2 . Then y^2 = 9 - x^2 ; y (distance from vertical midline to edge) = (9 - x^2)^0.5 ; 2y (the length of the horizontal slice) = 2*(9-x^2)^0.5 ;

So the integral is (ρg * 2 * x * (9-x^2)^0.5)dx , from 0 to 3 (feet). You should be able to handle this with the Chain Rule, right?

-- Cheers, -- Mr. d.