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 :

1] Dia. of each nozzle = 0.8 cm

= 0.008 m

2] Area of flow at each nozzle = 10 m/s

$\therefore$ Discharge through each nozzle

$Q = A \times V$

$= 0.00005026 \times 10$

$= 0.0005026 m^3/s$

Step No (1)

Torque exerted by water coming through nozzle A on the sprinkler = moment of momentum f water through A.

$= r_A \times \rho \times Q \times V_A$

$= 0.25 \times 1000 \times 0.0005026 \times 10$

Torque exerted by water coming through nozzle B on the sprinkler.

$= r_B \times \rho \times Q \times V_B$

$= 0.20 \times 1000 \times 0.0005026 \times 10$

$\therefore$ Total torque exerted by water on sprinkler.

$= 0.25 \times 1000 \times 0.0005026 \times 10 + 0.20 \times 1000 \times 0.0005026 \times 10$

= 1.2565 + 1.0052

= 2.26 N.m.

Step No (2)

Speed of rotation of arm, if free to rotate.

w = speed of rotation of sprinkler.

Absolute velocity of flow of water at the nozzle A and B are

$v_1 = 10 - 0.25 \times w$

$v_2 = 10 - 0.20 \times w$

Torque exerted by water coming out at A, on sprinkler.

$ = r_A \times \rho \times Q \times v_1$

$ = 0.25 \times 1000 \times 0.0005026 \times (10 - 0.25w)$

= 0.12565 (10 - 0.25 w)

Torque exerted by water coming out at B, on sprinkler.

$ = r_B \times \rho \times Q \times v_2$

$= 0.20 \times 1000 \times 0.0005026 \times (10 - 0.2w)$

= 0.10052 (10.0 - 0.2w)

$\therefore$ Total torque exerted by water

= 0.12565 (10.0 - 0.25w) + 0.10052 (10.0 - 0.2w)

Since moment of momentum of the flow entering is zero and no external torque is applied on sprinkler, so the resultant torque on the sprinkler must be zero.

$\therefore$ 0.12565 (10 - 0.25w) + 0.10052(10 - 0.2w) = 0

$\therefore$ 1.2565 - 0.0314w + 1.0052 - 0.0201 w = 0

$\therefore$ 1.2565 + 1.0052 = w (0.0314 + 0.0201)

$\therefore$ 2.2617 = 0.0515 w

$w = \frac{2.2617}{0.0515}$

w = 43.9 rad /s

and $N = \frac{60 \times w}{2 \pi }$

$= \frac{60 \times 43.9}{2 \pi }$

N = 419.2 r.p.m

A lawn sprinkler has two nozzles of diameters 8 mm each at the end of a rotating arm and the velocity of

flow of water from each leis 12 m/s. One azzle discharges water in the downward direction, while

the other mozzle discharges water vertically up. The nozzles are at a distance of 40 cm from the centre of the tating. Determine the worque required to hold the rotating arm stationary Also determine the con stant speed of rotation of arm, if it is free to rotate,