An airplane propeller is 2.08 m in length (from tip to tip) and h...

Question

An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airplane's engine is first started, it applies a constant torque of 1950 N•m to the propeller, which starts from rest. (e) What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 revolutions?

Accepted Answer

Welcome back everybody. We are making observations about a rod here. So let me go ahead and draw out our rod. We are told that one of the ends of this rod is attached to a Shaft of an electric motor. We're told a couple different things about this whole system here. We are told that the length of the Rod is 1. m and we are told that the mass is kg. Now we're told that the electric shaft supplies a uniform torque of 182 Newton m to the Rod when it is initially at rest. Now, we are tasked with finding what the instantaneous power is delivered to the rod at the moment that the rod completes eight revolutions. There's a lot of variables here, but let's just break it down. It all comes down to this. We need to find our instantaneous power, instantaneous power is simply equal to the torque times our final angular velocity. We have a torque but we've got to figure out this term right here. When I think of angular velocity and we are given an initial angular velocity as well as like a number of revolutions. We get to use this charismatic formula. Right here, we have our final angular velocity squared is equal to our initial angular velocity squared plus two times our angular acceleration times our change in data. We have everything here, except this angular accelerations. How do we go about finding that? Well, we know that our torque is equal to the moment of inertia times, or angular acceleration. But what is our moment of inertia? Or moment of inertia for a uniform rod? Where it's it's about one end is one third times the mass times the length squared. So what we gotta do, we gotta solve for the moment of inertia, solve for angular acceleration to solve for our final angular velocity. So let's just take it one step at a time. Here we have that our moment of inertia is going to be equal to one third times 73 times 1.36 squared. This is equal to 45 kg meters squared. Great. So now I'm gonna take a look at this formula here we have that our torque is equal to the moment of inertia times the angular acceleration. Let me go ahead and divide both sides by our moment of inertia. You'll see that on the right those terms cancel out. And we get our angular acceleration is equal to the torque divided by the moment of inertia. So we can now go ahead and calculate that We have that are angular acceleration is 182, which was our torque divided by our moment of inertia of 45, giving us 4.04 radiance per second squared. Now that we have found that we are ready to find our final angular velocity. Now, I'm just gonna take the square root of both sides of this equation. And you'll see that that gets rid of this power right here. So we have that our final angular velocity is equal to the square root of our initial angular velocity, which is zero squared plus two times our angular acceleration of four point oh four times our number of revolutions completed, which is eight revolutions. But in order to use this Kinnah Matic formula, we need to convert this to radiance. So let me do that real quick. We have that in one revolution, we have two pi radiance And then these units cancel out. And then when we just plug all this into your calculator, we'll get the right units. So, plugging all this in, we get that our final angular velocity is 20.15 radiance per second. Now that we have found our final angular velocity, we are ready to find our instantaneous power, instantaneous power at the time the eight revolutions are completed is going to be the torque times our final angular velocity equal to 182 times 20.15. Giving us 3668 joules per second. As you can see here, we need our final answer in watt. So let's see here, a jules per second. That unit is equivalent to one watt. These units will cancel out. And we get that our final instantaneous power is 3668 watts corresponding to our final answer. Choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

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