Density of holes on a sprinkler for uniform water distribution

[forestmine]

Hi all,

My classmates and I have been at this problem for some time, and it doesn't look like we're getting anywhere. We'd really love any help in the right direction!

Homework Statement

A lawn sprinkler is made from a spherical cap (max angle  = 45\)
with a large number of identical holes, with density n(theta). Determine n(theta) such that
the water is uniformly sprinkled over a circular area. The surface of the cap is level
with the lawn. Assume that the size of the cap is negligible compared to the size of
the lawn to be watered and neglect air resistance. Sketch your answer for n(theta).

Homework Equations

Area of a ring for a distance dr from the sprinkler: A = 2*pi*dr.

The Attempt at a Solution

Intuitively, I believe that if the angle is 0 from the tip of the sprinkler down to 45 at the ground, the density of holes ought to increase as you move towards the ground. Further, I would think that the ratio of the density of wholes to the area of a infinitesimal ring A should remain constant in order for the water to be uniformly distributed. I'm really not sure if we're headed in the right direction, though. Any help would be greatly appreciated!

 

[berkeman]

Hi all,

My classmates and I have been at this problem for some time, and it doesn't look like we're getting anywhere. We'd really love any help in the right direction!

Homework Statement

A lawn sprinkler is made from a spherical cap (max angle  = 45\)
with a large number of identical holes, with density n(theta). Determine n(theta) such that
the water is uniformly sprinkled over a circular area. The surface of the cap is level
with the lawn. Assume that the size of the cap is negligible compared to the size of
the lawn to be watered and neglect air resistance. Sketch your answer for n(theta).

Homework Equations

Area of a ring for a distance dr from the sprinkler: A = 2*pi*dr.

The Attempt at a Solution

Intuitively, I believe that if the angle is 0 from the tip of the sprinkler down to 45 at the ground, the density of holes ought to increase as you move towards the ground. Further, I would think that the ratio of the density of wholes to the area of a infinitesimal ring A should remain constant in order for the water to be uniformly distributed. I'm really not sure if we're headed in the right direction, though. Any help would be greatly appreciated!

(Thread moved from Advanced Physics to Intro Physics)

I'm not sure it helps, but I made a sketch of the sprinkler head and the initial trajectory angles necessary to hit 8 evenly spaced spots on the lawn, from the farthest out (45 degree launch) to the closest in (not including a hole for straight up). I think if you solve for the angles for a moderate number of trajectories, you will start to get a feel for the general function. Maybe give that a try?