corbell777
New member
- Joined
- Feb 21, 2012
- Messages
- 9
A hemispherical tank of radius R is filled with water which is pouring under the influence of gravity out of a circular hole of radius r at the bottom of the tank. Assume that the velocity of fluid flowing from the opening is proportional to sqrt(2gh), where g is gravity and h is the height of the water at time t. Derive a differential equation for the depth of the water at any time.
Change in height of fluid over time = Velocity of fluid leaving tank x area of opening
pi*R*R - pi*(R - h)*(R-h) dh = pi*r*r sqrt(2gh) dt
{(2Rh(to 1/2) - h(to 3/2) )/ r*r*sqrt(2g)} dh = dt
Integration gives
{(20R*h(to 3/2) - 6h(to 5/2) )/(15r*r*sqrt(2g)) = t
This is the answer which I got which matches the answer in the book. But it seems backwards to me.
If h = 0, then the tank is empty. And since it started full, there should be some expression for t. But if you put h = 0 in the above expression, you get t = 0.
Also, if R = h, then the tank is full, which should be at time t = 0, since the tank started full. But if you put R = h into the above expression, you get
t = (14R*h(to 5/2)/(15 r*r*sqrt(2g)).
So I think it should be the other way around. When R = h, I should get t = 0 and when h = 0, I should get the last expression for t.
Can someone explain to me what I'm getting wrong?
Change in height of fluid over time = Velocity of fluid leaving tank x area of opening
pi*R*R - pi*(R - h)*(R-h) dh = pi*r*r sqrt(2gh) dt
{(2Rh(to 1/2) - h(to 3/2) )/ r*r*sqrt(2g)} dh = dt
Integration gives
{(20R*h(to 3/2) - 6h(to 5/2) )/(15r*r*sqrt(2g)) = t
This is the answer which I got which matches the answer in the book. But it seems backwards to me.
If h = 0, then the tank is empty. And since it started full, there should be some expression for t. But if you put h = 0 in the above expression, you get t = 0.
Also, if R = h, then the tank is full, which should be at time t = 0, since the tank started full. But if you put R = h into the above expression, you get
t = (14R*h(to 5/2)/(15 r*r*sqrt(2g)).
So I think it should be the other way around. When R = h, I should get t = 0 and when h = 0, I should get the last expression for t.
Can someone explain to me what I'm getting wrong?
Last edited: