Problem Statement
The lawn sprinkler shown below has nozzles of 5mm diameter and carries a total discharge of 0.20 lps. Determine the angular speed of rotation of the sprinkler and torque required to hold the sprinkler stationary. Assume no friction at the pivot.
Given Data
Solution Approach
This problem involves the application of angular momentum principles. We’ll analyze the motion of the sprinkler by:
- Determining the relative and absolute velocities at each nozzle
- Applying the principle of angular momentum conservation
- Calculating the angular velocity for free rotation (zero net torque)
- Finding the torque required to hold the sprinkler stationary
Calculations
Part A: Angular Speed of Rotation
Step 1: Understanding the initial conditions
The initial moment of momentum of fluid entering the sprinkler is zero. Since no external torque acts on the system (no friction at pivot), the final moment of momentum should also be zero when the sprinkler rotates freely.
Due to the reaction forces at the nozzles, torque is generated:
- At nozzle 1: Anticlockwise torque
- At nozzle 2: Clockwise torque
Since the torque arm for nozzle 2 (r2) is greater than for nozzle 1 (r1), the sprinkler will rotate clockwise if free to rotate.
Step 2: Determining absolute velocities
The absolute velocity at each nozzle depends on the relative velocity and the tangential velocity due to rotation:
(Tangential velocity and relative velocity are in the same direction)
(Tangential velocity and relative velocity are in opposite directions)
Step 3: Setting up the angular momentum equation
For free rotation, the final net moment of momentum must be zero:
Since ρ and Q are the same for both nozzles, we can simplify:
Step 4: Solving for angular velocity (ω)
Step 5: Converting to RPM
Angular Speed (ω) = 10.18 rad/s = 98 rpm
Part B: Torque Required to Hold Sprinkler Stationary
Step 1: Setting up the torque equation
When the sprinkler is stationary (ω = 0), the absolute velocities equal the relative velocities (V1 = V2 = 5.09 m/s). The torque exerted by water on the sprinkler is:
Step 2: Calculating the torque
Torque Required (T) = 0.0509 N·m
Detailed Explanation
Working Principle of Lawn Sprinklers
Lawn sprinklers operate based on the principle of action and reaction (Newton’s Third Law). As water jets out from the nozzles, it creates a reaction force in the opposite direction, causing the sprinkler to rotate.
Conservation of Angular Momentum
In this problem, we applied the principle of conservation of angular momentum. When the sprinkler rotates freely, the total moment of momentum of the system remains constant. Since the initial moment of momentum is zero (water entering axially), the final moment of momentum must also be zero when the sprinkler reaches steady-state rotation.
Absolute vs. Relative Velocity
An important aspect of this problem is understanding the difference between relative and absolute velocities:
- Relative velocity: The velocity of water relative to the nozzle (V = Q/A)
- Absolute velocity: The velocity of water relative to a stationary observer
When the sprinkler rotates, the absolute velocity is affected by the tangential velocity of the nozzles (r·ω). For nozzle 1, these velocities add up, while for nozzle 2, they partially cancel each other.
Torque Analysis
The torque exerted by the water on the sprinkler is proportional to:
- The mass flow rate (ρQ)
- The absolute velocity of the water
- The moment arm (radius) from the pivot
When the sprinkler is held stationary, the torque represents the force required to prevent rotation. This torque (0.0509 N·m) is relatively small, which is typical for garden sprinklers.
Practical Applications
This analysis has several practical applications:
- Design of efficient irrigation systems
- Optimization of water distribution patterns
- Calculation of sprinkler coverage areas
- Understanding of reaction turbine principles (similar to certain hydro turbines)
Engineering Considerations
In real-world sprinkler design, several additional factors would be considered:
- Frictional losses at the pivot
- Air resistance on the rotating arms
- Pressure variations in the water supply
- Optimal rotation speed for desired coverage pattern
- Material selection for durability and efficiency
Analysis of Results
The calculated angular speed of 98 rpm (or 10.18 rad/s) is typical for lawn sprinklers. At this speed, the sprinkler provides effective water distribution while maintaining structural integrity. The calculated torque of 0.0509 N·m indicates the force needed to hold the sprinkler stationary, which is useful in designing mounting mechanisms and testing equipment.
