A 10 g bullet traveling at 400 m/s strikes a 10 kg, 1.0-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door just after impact?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A student throws a 5.0 g tip in the X direction with a speed of 25 m per second toward a square shaped vertical swinging plate made of foam attached to a pivot. At one end, the plate has a mass of 125 g and a side of 80 centimeters, the tip is implanted in the foam five centimeters from the opposite side of the pivot, calculate the angular speed of the plate just after collision. So that is our end goal is to find the angular speed of the plate just after the collision. So we're given some multiple choice answers. They're all in the same units of gradients per second. So let's read them off to see what our final answer might be. A is 1.1 B is 2.1 C is 3.2 and D is 3.8. OK. So first off to help us visualize this problem. Let's look at a diagram. So here is a handy dandy diagram to help us visualize this problem. It shows the pivot. It also shows the tip and the direction of the tip is facing towards the 80 centimeter side of the plate. In this 80 centimeters, they wrote it in meters. So it's 0.8 m and this 75. So 0.75 m is the distance from the pivot, which we'll talk about that more in a second. OK. So let us model the tip as a particle. Also let us note that we're dealing with a collision that results in a change of in rotation. So we can conserve angular momentum. In this case, the equation for the initial angular momentum, we're gonna call it L subscript IL I with respect to the pivot can be written as L I which is the initial angular momentum is equal to the mass of the tip multiplied by our tip, which our tip is the distance from the pivot, which is the 75 centimeters I was mentioning before or the 0. m multiplied by V tip, the velocity of the tip multiplied by sine of degrees. So sine of 90 degrees is one. So then we can simplify this to say that the initial angular momentum is equal to the mass of the tip multiplied by our tip multiplied by V tip. OK. So the equation for the final angular momentum, we're gonna call it L subscript FLF with respect to the pivot can be written as LF. So the final angular momentum is equal to I total multiplied by Omega F where I total is the total moment of inertia. And Omega F is the angular speed. OK. So the equation for the total moment of inertia with respect to the pivot is I total equal to I plate plus I tip. So I total is equal to one third multiplied by the mass of the plate. I'm gonna call it MP multiplied by A squared plus M tip multiplied by R tip squared where A is the width of the plate. In this case, it's 80 centimeters. OK. So applying the conservation of momentum rules, we can write that the initial angular momentum is equal to the final angular momentum. So we can say that the mass of the tip multiplied by our tip multiplied by V tip is equal to one third multiplied by the mass of the plate multiplied by A squared plus M tip multiplied by R tip squared multiplied by the angular frequency, the final angular frequency. So now we need to rearrange this equation to solve for the angular frequency. So let's do that or the final angular frequency, we should be specific. So the final angular frequency, when we use a little bit of algebra to isolate and just solve for the final angular frequency, we will get the it is equal to M tip. So the mass of the tip and tip multiplied by R tip multiplied by V tip, divided by one third multiplied by the mass of the plate. MP multiplied by A squared plus M tip multiplied by R tip squared. So now we can substitute all of our known variables to solve for the numeric numerical value of the final angular frequency. So the final angular frequency or I should say angular speed is equal to the mass of the tip, which was 5.0 g. We need to convert grams to kilograms. So we need to multiply it by 10. The power of negative three multiplied by our tip, which we know that to be 0.75 m as we mentioned before, multiplied by V tip which the speed was 25 m per second. Awesome divided by one third multiplied by the mass of the plate which was 125 grams. So you need to convert that to kilograms. So 125 multiplied by 10 to the power of negative three, convert it to kilograms grams to kilograms multiplied by a squared which was a is the width of the plate which is 80 centimeters, but we need to convert 80 centimeters to meters which is 0.8 meters. So we convert centimeters to meters to be clear square plus the mass of the tip which was 5.0 multiplied by 10 to the power of negative 3 kg multiplied by R tip squared, which was 0.75 m squared. OK. So when we plug that into a calculator, we should get that the final angular frequency is equal to 3.2 radiance per second, which is our final answer. We did it. So just after the collision, the angular speed of the plate is 3. radiance per second. Awesome. So let's look at our multiple choice answers to see what our final answer has to be. The correct answer is the letter C 3.2 radiant per second. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.