A uniform, 4.5-kg, square, solid wooden gate 1.5 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1-kg raven flying horizontally at 5.0 m/s flies into this door at its center and bounces back at 2.0 m/s in the opposite direction. (a) What is the angular speed of the gate just after it is struck by the unfortunate raven?

Question

Accepted Answer

everyone welcome back in this problem. We have a uniform window okay. With some dimensions and math, it is pivoted by frictionless hinges okay along its width and it hangs vertically and we have this poor peregrine falcon. This bird okay? With a speed is going to hit the window at its center. The falcon is going to bounce back. And what we want to know is what is the windows angular speed immediately after it is hit by that bird. Okay, so let's go ahead and just draw a diagram of what's going on in this problem. Okay, So we're going to take right to be the positive extraction. And let's start by drawing out the window. Okay? So we're told that it's .7 m long and .5 m wide. Okay, so drawing that out. It's going to look kind of like this. So the length 0.7 m. Let me just shorten that arrow off of it. Okay, And the width Is 0.5 m. Alright, and we're told that it has a massive 11 kg. Okay, so 11 kilograms. Now we're told that it's on frictionless hinges along its width. So the hinges will be along the width here and that it's hanging from those hinges vertically. All right, now we have our bird. So this is our falcon and he's going towards the window. He's 0.85 kg. We're told that he has a speed. Okay, And this is a linear speed because he's not going in circles like he's not got angular velocity, that's a linear, so he's flying At 100 km an hour. And because we have them going to the left and we've taken right to be the positive X direction. This means that our bird will have a negative velocity. Alright? And we're told that he's going to hit the center, so he's going to go and hit right here. All right now, after he hits that window, we are left with this situation. Okay, So this is before the collision and this is after the collision we have the same window, it's still on its hinges. It's the same size as before, same width in height and mass. And now we're told that our falcon Again, kg with the flight speed of 200 km an hour before he's bouncing back at 60 km/h. So he's going this way And his velocity is going to be 60 km an hour and this time it's positive because he's going to the right to the positive X direction. All right, great. So, what we want to know is what is the windows angular speed immediately after it is hit. Okay, So the window is going to rotate when that bird hits the window with some speed. We'll call it omega of the window final. Okay. And we want to know what is that value. Alright. We have a collision problem like this. What can we use We know we have conservation of angular momentum and we know this because we have no net external torque. Okay. Our hinges are frictionless. We're told that in the problem. Okay, so we have no net external torque. So we're going to have a conservation of angular momentum. Conservation of angular momentum tells us that the angular momentum initially l not is going to equal the final angular momentum. L. F. Alright, what components do we have in our system? Okay, this is the angular momentum of the whole system. What do we have to consider in our system? Well, initially we have the window and the bird and after the collision, same thing we have the window and the birds. So for each of these we need to consider the angular momentum of the window plus the angular momentum of the bird. So we have L Oh my oh sorry, angular momentum L. Of the window initially plus the angular momentum of the bird initially. Okay. And that's going to equal the angular momentum of the window, final plus the angular momentum of the bird. Alright, now let's recall when we're talking about angular momentum, we have I the moment of inertia times, omega, the angular velocity of the angular speed. So for each of these terms we're gonna have I make a knot or w not and then omega w not. Alright. When we're talking about the bird, the bird does not have angular velocity, it is flying straight, it has linear velocity. And so when we are dealing with angular momentum of something with linear motion, the angular momentum recall is going to be written as M R. B. Signed data. Okay, so in this case we're going to m the mass of the bird. Are the radius of the bird initially Mhm B The velocity of the bird initially. Time sign the theta. And welcome back to what that sine of theta is. And that's going to equal the final. Okay, so again, the window has angular speed so it's going to have angular velocity given by or sorry, angular momentum given by I times omega. Okay, so, I of the window final omega of the window final. Okay, and the bird again, that's linear motion. So that's going to be M R. V. Signed data, mass of the bird really is the lost City. Alright, great. Now working this out on the left hand side, we are going to get I omega not of the window. Okay, well, the window isn't moving initially, it doesn't have angular motion or it doesn't have angular velocity or speed. And so this term is just going to go to zero. And so we're going to get just the mass of the bird 0.85 kg. This radius. Okay, and this is the radius from the axis of rotation. Okay. And we're told the bird is hitting in the center and so That's gonna be halfway down. So this is gonna be a distance of 0.535 m. The entire decision .7 m. So halfway is going to be .35 m. Okay, so the radius is going to be 0.35 m. The velocity when we know the velocity is -100 km/h. Let's go ahead and just write that in terms of meters per second. Okay, we need consistent units, so we have 100 km/h, We can multiply that by one hour And that's gonna be 3600 seconds And we can also multiply by m/km2. And if we do this, the kilometers will cancel the hours will cancel and we are going to be left with meters per second, the unit that we want in this case it's going to be 27. m per second. Okay, Now, when we're putting this in here, remember we're talking about momentum direction matters, we're using velocity, so we need the direction. So in this case it's going to be negative 27.78 m per second, kid to indicate that it's going to the left and then sign data data is going to be the angle that it makes okay with the axis. And so because we're hitting the window right in the center, You can imagine it flying into the window. Okay. And so the angle that's gonna be a perpendicular collision and so the angle is going to be 90°. Okay, sign of 90°. All right, on the right hand side. The moment of inertia of the window finally. Alright, well, let's work that out before we plug in here. Okay, So we want to find I the moment of inertia of the window finally. Okay, well, our window is a slab, like a rectangular slab. Okay, So the moment of inertia recall. Use your table. Look at the moment of inertia table that is either in your text worker that you've been provided with by your professor. Okay. And that's gonna be one third M H squared. In this case we have one third, the mass of the window is 11 kg and then the height while it's going to be .7. Okay, so 0.7 squared. Alright. And that's gonna give us a moment of inertia of 1. kg meters squared. All right. So, let's use that in our equation here. So, the moment of inertia of the window after the collision, 1.7967 m per second. Whoops. Sorry, kilogram meters squared. This is why it's important to write those units out. So you don't make any mistakes, kilogram meters squared. Ok. Omega of the window. Final Well, that's what we're trying to find out. Okay, that's what we want to know. What is that angular speed of the window after the bird runs into it. Alright. And then in terms of the bird, It has the same mass as before 0.85 kg. The radius is going to be the same. Okay, the velocity. Well, we're told that 60 km/h, So 60 km/h. Eight times same as before. So, we're timing by one over 3600 and then 1000 over 3600 K. And this is going to be our minute per hour hour meter per oh kilometer second. All right, so we end up with A speed of 16.67 m/s. K 16.67 m per second. And then at the end, we're going to multiply by sine of 90 degrees. Okay, times sine of 90°. And we ran out of room a little bit there. So just written that below. Alright, let's roll scroll down so we can keep working. Alright, so working out some of these details, we get -8.26455. Okay, And we have a unit of kilogram Okay, then we times my meter that many times my meter per second. We have kilogram meter squared per second, left hand side, 1.7967 kgm squared. Omega WF plus 4.9593 - five. Alright, And here, same thing, kilogram meters squared per second. Alright, so if we move this over to the left hand side, we're gonna have -13. three, not 8223875 on the right hand side with 1.7967 omega WF. Alright. And dividing? And again we have kilogram meters per second, kilogram meters per second. We have the sign 90. Um and then we have Yeah, so we have the per second. Okay. So the kilogram meters squared are going to be canceled when we divide and we are going to be left with omega W. F equals 7. radian per second. So that is the angular speed of the window after the bird hits it. Okay, So that is going to correspond with answer C 7.37 radiance per second. Okay. I hope this video helped. Thanks everyone for watching.

Verified Solution

Key Concepts

Conservation of Angular Momentum

Moment of Inertia

Angular Speed