/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 131 \( \mathrm{A}\) jet of oil \((\m... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 131

\( \mathrm{A}\) jet of oil \((\mathrm{SG}-0.8)\) strikes a curved blade that turns the fluid through angle \(\theta-180^{\circ}\). The jet area is \(1200 \mathrm{mm}^{2}\) and its speed relative to the stationary norzle is \(20 \mathrm{m} / \mathrm{s}\). The blade moves toward the nozzle at \(10 \mathrm{m} / \mathrm{s}\). Determine the force that must be applied to maintain the blade speed constant.

Short Answer

Expert verified
The force that must be applied to maintain the blade speed constant is 192 N.

Step by step solution

01

Calculate the Relative Speed of the Oil Jet

Firstly, find the relative speed of the oil jet with respect to the blade. The relative speed can be calculated by subtracting the speed of the blade from the speed of the oil jet. The speed of the oil jet relative to the stationary nozzle is given as 20 m/s and the speed of the blade is given as 10 m/s. Therefore, the relative speed of the oil jet with respect to the blade would be 20 m/s - 10 m/s = 10 m/s.
02

Calculate the Mass Flow Rate

The next step involves calculating the mass flow rate (the amount of mass flowing per unit time). The mass flow rate can be calculated by multiplying the density of the fluid (oil in this case), the cross-sectional area through which it flows and its relative speed. Since the specific gravity (SG) of oil is given as 0.8, we can calculate the density by multiplying the specific gravity with the density of water (which is 1000 kg/m鲁). Therefore, the density of oil is 0.8 * 1000 kg/m鲁 = 800 kg/m鲁. Also, convert the jet area from mm虏 to m虏 by multiplying by \(10^{-6}\). Therefore, the jet area is 1200 mm虏 * \(10^{-6}\) m虏/mm虏 = 0.0012 m虏. Now the mass flow rate can be calculated as follows: Mass Flow Rate = Density * Area * Velocity = 800 kg/m鲁 * 0.0012 m虏 * 10 m/s = 9.6 kg/s.
03

Calculate the Force

To maintain the blade speed constant, a certain force must be applied. The change in momentum due to the jet is given by the mass flow rate multiplied by the velocity component. Since the fluid turns through 180 degrees, the momentum change is doubled, implying the fluid comes to a stop and is then forced to move in the reverse direction with the same speed. The force required to affect this momentum change per unit time is the rate of change of momentum, which is the same mass flow rate multiplied by the change in velocity. Therefore, Force = Mass Flow Rate * Velocity Change = 9.6 kg/s * 2 * 10 m/s = 192 N.

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