A fire hose exerts a force on the person holding it due to the...

Question

A fire hose exerts a force on the person holding it due to the water accelerating as it goes from the thicker hose out through the narrow nozzle. How much force is required to hold a 7.0-cm-diameter hose delivering 480 L/min through a 0.75-cm-diameter nozzle?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A garden water sprinkler rotates and ejects water through a small nozzle. The sprinkler is connected to a hose that has a diameter of 2.0 centimeters and is designed to deliver water at a rate of 300 L per hour. The water exits the sprinkler through a nozzle that has a diameter of 0.50 centimeters. How much force is exerted on the sprinkler due to the water flowing from the hose through the nozzle. So that's our goal. Our angles are trying to figure out how much force is exerted on the sprinkler due to this water flowing through the hole through this hose through the nozzle. So that's our angle. We're ultimately trying to figure out what the force exerted on the sprinkler is due to the conditions provided to us by the pro itself. So now that we're trying now that we know that we're trying to solve for the force exerted on the sprinkler. Let's read off multiple choice answers to see what our final answer might be noting that they're all gonna be in the same units of newtons. So A is 0.28 B is 0.33 C is 0.43 and D is 0.64. Ok. So first off, let us write down all of our known variables. So we know the diameter of the hose. Let's call this D one and D one is equal to 2.0 centimeters, which is also equal to 0.020 m. So to quickly pause here, I've gone ahead and done the conversion for you to save some time. But you can use dimension analysis to convert centimeters to meters or you could just quickly look it up yourself. We also know the diameter of the nozzle, which we call this D two and D two. The diameter of the nozzle is equal to 0.50 centimeters, which is also equal to zero 0.0050 m. Awesome. We also know the flow rate which we're gonna call the flow rate. Q and the flow rate is equal to 300 L per hour, which this is an example of using dimensional analysis is we need to convert liters per hour to cubic meters per second. So we need to take 300 L per hour and we need to recall that in one hour there is 3600 seconds. And we need to recall that in one cubic meter, there is 1000 L. So when we plug that into our calculator and we round to two decimal places, we will get 8.33 multiply by 10 to the power of negative five. And its new units of course will be cubic meters per second. Fantastic. So now we must note that for this particular problem, we will be dealing with the magnitude of the force only. We must also recall that the continuity equation states that the flow rate is constant. So with that in mind, we must also recall and use the continuity equation which states that Q is equal to V divided by T, which is also equal to a one multiplied by V one which is equal to A two multiplied by V two. So we need to note that a one and V one are the cross sectional areas of the hose and the water speed in the hose respectfully. And A two and V two are the cross sectional areas of the nozzle and the water speed in the nozzle respectfully. So now at this point, we must solve for a one and a two by recalling and using the area of a circle formula considering both the hose and the nozzle conditions separately. So for the hose, a one for the hose, a one value we can use the circular formula like. So we could say that A one is equal to pi multiplied by D one divided by two all to the power of two, which is equal to when we plug in our known variables, pi multiplied by 0.02 m divided by two all to the power of two, which is equal to when we round to two decimal places, 3.14 multiplied by 10 to the power of negative 4 m squared. And for the nozzle, a two value, we could write that A two is equal to pi multiplied by D two divided by two all to the power of two which playing are known variables will give us pi multiplied by 0.0050 m divided by two all to the power of two which is equal to once again rounding the two decimal places 1.96 multiplied by 10 to the power of negative five and its units are meters squared. So now at this point, we must solve for the speed in terms of the flow rate by recalling and using the following equations to help us solve for V one and V two. Let us recall that V one is equal to Q divided by A one which is equal to plugging in our known variables. 8.33 multiplied by 10 to the power of negative five. And don't forget that the units are meters cubed per second divided by a one which we just found a moment ago to be 3.14 multiplied by 10 to the power of negative 4 m squared, which when we plug it into a calculator that will give us 0.265 m per second. And similarly, for V two, V two is equal to Q divided by A two. So plugging in are known variables once again. So 8.33 multiplied by 10 to the power of negative five cubic meters per second, divided by our A two value which is one 0.96 multiplied by 10 to the power of negative 5 m squared. And when we plug that into our calculator, we will get 4.25 when we round to two decimal places, meters per second. So now at this point, we must recall that according to Newton's second law, the equation to solve for the force f exerted by the water by the nozzle can be written as F equals where F is the force M multiplied by delta V divided by delta T which is equal to row multiplied by Q multiplied by V two minus V one. Where let's make a little side note off to the side in blue where row in this case is the density of water and the density of water is equal to 1000 kilograms per meters. Cute. Awesome. So with that in mind, now we can plug in all of our known variables to solve for F which is our final answer that we're ultimately trying to solve for. So F is equal to row which is 1000 kilograms per cubic meter. So kilograms per cubic meter. And we need to multiply this value by Q which is once again 8.33. And we need to multiply it by 10 to the power of negative five and then its units are meters cubed per second. And now we need to multiply this by 4.25 m per second. Mine is zero 0.265 meters per second. Awesome, because that's our V one value that we rounded the three decimal places. So let me plug that into our calculator. We will find that fr force is equal to 0.3319 newtons, which is approximately equal to 0.33 when we round to two decimal places and its units are newtons and that's it we've saw for our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter B 0.33 newtons. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

A fire hose exerts a force on the person holding it due to the water accelerating as it goes from the thicker hose out through the narrow nozzle. How much force is required to hold a 7.0-cm-diameter hose delivering 480 L/min through a 0.75-cm-diameter nozzle?

Key Concepts

Continuity Equation

Bernoulli's Principle

Force Calculation

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