Project ForgeA sprinkler rotates about its hinge point in a horizontal plane due to water flow discharged through its two exit nozzles, at radial distances of 10 cm and 20 cm from the hinge (both nozzles ejecting in the same absolute tangential sense, on a straight arm through the hinge). The total flow rate Q through the sprinkler is 1 litre/sec and the cross-sectional area of each exit nozzle is 1 cm². Assuming equal flow rate through both arms and a frictionless hinge, the steady state angular speed of rotation (in rad/s) of the sprinkler is ______ (correct to two decimal places).
Explanation
Each arm gets Q/2, so relative exit velocity vrel = (Q/2)/A = 500 cm³/s / 1 cm² = 500 cm/s = 5 m/s at both nozzles. With a frictionless hinge and no external torque, the moment-of-momentum balance about the hinge requires the two arms' angular-momentum-flux contributions to be equal (since the arm is straight through the hinge, one nozzle's absolute tangential speed is vrel+ωrA while the other's is vrel−ωrB): ṁA·rA·(vrel+ωrA) = ṁB·rB·(vrel−ωrB). Equal mass flow rates cancel: 0.1(5+0.1ω) = 0.2(5−0.2ω) → 0.5+0.01ω = 1−0.04ω → 0.05ω=0.5 → ω=10 rad/s.
Source: GATE ME 2018 Session 1 (Set 1) (2018)
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