Force of Wind The force of the wind blowing o...

Question

Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft², how much force will a wind of 80 mph place on a surface of 2 ft²?

Accepted Answer

Hey, everyone in this problem, we have a truck carrying an aluminum sheet tied in the vertical position. The sheet is experiencing a wind force which is jointly proportional to the surface area of the sheet and the square of the wind's velocity. We're asked to find the amount of force experienced by the sheet for a wind velocity of 40 MPH in a surface area of four ft squared. We're told that the sheet experienced a force of £25 for the surface area of one ft squared in a wind velocity of 10 MPH. We're given four answer choices, option, a £1600 option B £16,000 option C £160 and option D £600. Now, we're told that we have this force, this wind force that is jointly proportional to the surface area of the sheet and the square of the wind's velocity. Ok. So the force is going to be proportional to both of these things. If it's jointly proportional, that means we're multiplying them together. Ok. So it's proportional to the surface area which we'll call a and also to the square of the wind's velocity, which we'll call V. Now, if we're talking about the square of the velocity, we have V squared. OK. So we have F is proportional to a multiplied by V squared. Now, we wanna turn this into aqua an equation so we can take that proportional symbol and write it as an equal sign if we introduce our constant. OK. So we have F is equal to some constant K multiplied by A B squared. OK. So we have this proportionality constant K there. And we've changed this into an equal sign. Now, what we wanna figure out is the force F, if we have a particular velocity in a particular surface area. Now, in order to find F, we need to know K knowing the surface area and the velocity A and B is not enough, we'll still have two unknowns in our equation. So let's use the other information we're given in this problem first to figure out what K is. And then we can use that in our equation to figure out the force for those particular um variables we were given. So we're told that when we have a force F of £25 we have a surface area a of one ft squared and a win the velocity V of 10 MPH. OK. So if we substitute this into the equation that we wrote, we have the, the force F £ is equal to that constant K multiplied by the area, one ft squared, multiplied by the velocity 10 MPH, all squared. OK. Solving for K, we get that K is equal to £25 divided by, in terms of the number here, we have one multiplied by 10 squared and it gives us in our unit, we have foot squared and then we have MPH squared. We're gonna divide by 100 multiplied by foot squared, multiplied by mile per hour squared. And we can simplify this. We have 25 divided by 100. Both of those are divisible by 25. And this is gonna give us one quarter in that same unit pounds per but squared mile per hour squared. Ok? So we figured out the value of K, now we can use that in our equation or F, OK? We now know that F is gonna be equal to one quarter A the square. OK. I've left out the units here just to make that um simpler and clearer and nicer looking here. But we're gonna use it um, the units when we do the calculation. OK. So doing that, we're trying to find the force F and we're trying to find the force F when we have a velocity of 40 MPH in an area of four ft squared. So we have F is equal to 1/4, £8 per foot squared mile per hour squared. That's our K value multiplied by the area, four ft squared, multiplied by the velocity, 40 MPH, all squared. Now, our units here are going to cancel. So the foot squared is gonna divide out the foot squared and the MPH with the MPH. And that's why it's important to include those units in that constant K. So we can see that they all divide out and we're left with just a unit of pounds. So we get that our force is equal to 1/4, multiplied by four. OK. That's gonna be just one multiplied by 40 squared. And so we get £ and, and that is that force we were looking for the force experienced by the sheet for that particular velocity and surface area. And that corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft², how much force will a wind of 80 mph place on a surface of 2 ft²?

Key Concepts

Joint Variation

Solving for the Constant of Variation

Applying the Variation Formula to New Conditions

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