Density of holes on a sprinkler for uniform water distribution

In summary, a group of students are seeking help with determining the density of holes on a spherical cap lawn sprinkler in order to achieve uniform water distribution over a circular area. They have attempted to use the area of a ring and have made a sketch to guide their approach, but are unsure if they are on the right track. They are seeking additional guidance and suggestions from the community.
  • #1
forestmine
203
0
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!
 
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  • #2
forestmine said:
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?
 

What is the purpose of the density of holes on a sprinkler for uniform water distribution?

The density of holes on a sprinkler is designed to ensure that water is distributed evenly over a specific area. This helps to prevent dry spots and over-watering in a lawn or garden.

How is the density of holes determined for a sprinkler?

The density of holes is typically determined by the manufacturer based on the size and shape of the sprinkler head, as well as the desired coverage area. It is important to follow the manufacturer's recommendations for optimal water distribution.

Can the density of holes on a sprinkler be adjusted?

Some sprinklers may have adjustable settings for the density of holes, allowing for more or less water to be dispersed. However, it is important to note that altering the density of holes may affect the uniformity of water distribution.

How does the density of holes affect water usage?

The density of holes on a sprinkler can have a significant impact on water usage. A higher density of holes may result in more water being used, while a lower density may result in less water being used. It is important to find a balance between water conservation and effective watering.

What happens if the density of holes on a sprinkler is too high or too low?

If the density of holes is too high, it may result in over-watering and potential water waste. On the other hand, if the density of holes is too low, it may result in dry spots and inadequate water coverage. It is important to find the optimal density of holes for your specific sprinkler and watering needs.

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