Problem Statement
A lawn sprinkler shown in the figure has 0.8cm diameter nozzle at the end of a rotating arm and discharges water at the rate of 12m/s. Determine the torque required to hold the rotating arm stationary. Also determine the constant speed of rotation of the arm, if free to rotate.
Given Data
Solution Approach
To solve this problem, we’ll apply the principles of angular momentum and jet propulsion. First, we’ll calculate the torque required to hold the rotating arm stationary, and then determine the constant speed of rotation when no external torque is applied.
Calculations
Part A: Torque Required to Hold Arm Stationary
Step 1: When the angular velocity (ω) = 0, the arm is stationary. The water jets at nozzles A and B create forces in opposite directions (upward at A and downward at B).
Both jets create torques in the same direction (clockwise), so the net torque is the sum of these torques.
Step 2: The torque exerted by each water jet is calculated using the momentum equation:
Where:
ρ = density of water (1000 kg/m3)
Q = discharge rate (0.000603 m3/s)
V = velocity of water (12 m/s)
r = arm length (0.2 m)
Step 3: Calculate the total torque:
Torque required to hold the rotating arm stationary = 2.89 Nm
Part B: Speed of Rotation when Free to Rotate
Step 1: When the arm is free to rotate, no external torque acts on the system (T = 0). According to conservation of angular momentum, the final moment of momentum should be zero.
The initial moment of momentum of fluid entering the sprinkler is zero, so the final moment of momentum must also be zero.
Step 2: Calculate the absolute velocities at nozzles A and B:
Note: The tangential velocity and relative velocity are in opposite directions.
Step 3: Set the final moment of momentum to zero:
Step 4: Convert angular velocity to rotational speed:
Constant speed of rotation of the arm = 573 rpm
Detailed Explanation
Principles of Jet Propulsion in Rotating Systems
This problem demonstrates the principles of jet propulsion and angular momentum in a rotating system. The lawn sprinkler operates on Newton’s Third Law of Motion – as water is expelled from the nozzles, a reactive force is created in the opposite direction.
Torque and Angular Momentum Analysis
When the arm is held stationary, the water jets create torques that tend to rotate the arm. These torques must be counterbalanced by an external torque to maintain the arm’s position. The calculation shows that a torque of 2.89 Nm is required.
Self-Regulating Rotation Speed
When the arm is free to rotate, it reaches a constant angular velocity. This happens because as the rotation speed increases, the absolute velocity of the water jets decreases (relative to a fixed reference frame). When the absolute velocity reaches a point where the angular momentum is balanced, the arm maintains a constant speed of 573 rpm.
Practical Applications
The principles demonstrated in this problem have practical applications in:
- Design of irrigation sprinklers
- Rotating nozzle systems in firefighting equipment
- Reaction turbines and propulsion systems
- Rotary lawn sprinklers for efficient water distribution
Energy Considerations
In a free-rotating sprinkler, part of the water’s energy is converted into rotational kinetic energy of the arm. This reduces the effective range of the water jet compared to a fixed nozzle but provides better area coverage through rotation.
Factors Affecting Performance
Several factors can affect the performance of a rotating sprinkler:
- Water pressure (affecting discharge velocity)
- Nozzle diameter and shape
- Arm length (longer arms create greater torque)
- Friction in the rotating mechanism
- Number and orientation of nozzles
Understanding these principles allows engineers to design efficient water distribution systems for irrigation, firefighting, and other applications where controlled fluid distribution is required.
