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Problem 52 Hard Difficulty

mmh The helicopter in the drawing is moving horizontally to the right at a constant velocity $\vec{v}$ . The weight of the helicopter is $W=53800 \mathrm{N}$ . The lift force $\vec{L}$ generated by the rotating blade makes an angle of $21.0^{\circ}$ with respect to the vertical. (a) What is the magnitude of the lift force? (b) Determine the magnitude of the air resistance $\vec{R}$ that opposes the motion.

Answer

$5.76 \times 10^{4} \mathrm{N}$ $\\$
$2.06 \times 10^{4} \mathrm{N}$

Discussion

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MH

Mallory H.

April 7, 2021

A worker stands still on a roof sloped at an angle of 36° above the horizontal. He is prevented from slipping by a static frictional force of 390 N.

MH

Mallory H.

April 7, 2021

please ignore below.

Video Transcript

we begin this question by choosing our frame of reference. So are you. Choose that everything that is pointing upwards on the Y direction is positive on everything that is pointing to the right on the X direction is positive, too. Then, for the first item, we have to apply Newton's second law on the vertical axis. So on the vertical axis, Newton's second law tells us that the net force is equals to the mass of the helicopter. Times acceleration off the helicopter, that direction. But the helicopter is moving on the horizontal, meaning that it's not moving on the vertical direction. Therefore, acceleration is it close to zero in that direction, Then the net force in the Y direction is equals to zero, and the net force in the Y direction is composed by the weight force, which is negative in our reference frame. And a portion off that lift force L on the portion that composes the vertical part is these one we shall call l. Why so the y component off the lyft force and this is the X component off the same force. Then Newton's second low tells us that the lift force in the y direction. Miners. The weak force is in course to zero. The reform The Lyft force in the Y direction is equal to the weight force which is equals to 53 800 new terms. Now, what is the relation between the Why component off the lift force and the magnitude off the food lift force? Note that we have a rectangle triangle using our components. These high punter news is the food lift force. These is the X component and this is the white component off that force and this angle is 21 degrees. By calculating the co sign off these angle, we can get our relation. The co sign off 21 Greece is equal to the address inside off that angle. So l component why divided by the hypotenuse l therefore l why is he course to L A times the co sign off 21 degrees, therefore l a times The cool sign off 21 degrees is equals to 53 800 Therefore, the magnitude off the lift force is it close to 53 800 divided by the co sign off 21 degrees. And these gives us approximately 57,000 and 600 new terms. So this is the magnitude off the force now. To detriment the magnitude off the resistance force, we have to apply Newton's second law on the horizontal access. Serve on the horizontal axis. You don't Second law tells us that the net force in that direction is equal to the mass off the helicopter times acceleration off the helicopter on the horizontal direction. Now the helicopter is moving on the horizontal, but it's moving with a constant velocity. Therefore, acceleration is he goes to zero in that direction. Truth. Then the net force in the X direction is in close to zero, but the net force is composed by two forces. The lift force X component and the resistant forced are. Then we have that the lift force X component, which is pointing to the positive direction off the horizontal axis minus. The resistant force is imposed 20 then ale Component X is equal to the resistant force. But what is L component X? Well, we can repeat this idea to get it. The side off 21 degrees is equals to the opposite side off the triangle. L X divided by the hypotenuse L Therefore l X is equals to l. A times this sign off 21 degrees. Then the resistant force is he goes to l times this sign off 21 degrees. But you know that l is equals to 57. 600. Then we have 57 600 times this sign off 21 degrees here, and these is approximately 20,000 and 600 new terms. These is the answer off the second item.