(II) A particle is at the position (x, y, z) = (1.0, 2.0, 3.0)...

Question

(II) A particle is at the position (x, y, z) = (1.0, 2.0, 3.0)m. It is traveling with a vector velocity (-5.0 ,+ 2.8, -3.1)m/s . Its mass is 4.3 kg. What is its vector angular momentum about the origin?

Accepted Answer

Hey, everyone. In this problem, a block is on a merry go round. Its position is given by XYZ is equal to 34 and 5 m. And the velocity can be expressed as negative four comma 3.2 comma negative 3.8 m per second. If the block is 5.2 kg or asked to determine its angular momentum with respect to the origin, we're given four answer choices at all in kilogram meters squared per second. Option, A negative 31 I hat plus 8.6 J hat plus 26 K hat. Option B negative 31 I hat minus 8.6 J hat plus 26 K hat. Option C negative 160 I hat plus 45 J hat plus 100 and 30 K hat. And option D negative 160 I hat minus 45 J hat plus 130 K hat. OK. So we're looking for the angular momentum and we call it the angular momentum L is equal to the moment of inertia. I multiplied by the angular velocity omega. And we can also write it as the cross product between the position vector R and the linear momentum P. Yes, we have these two different ways to calculate it. Now, we're given a position vector R and so we already know that R is equal 23 I hat plus four J hat plus five K hat in meters. OK. So we were given that we're just writing it in terms of the IJ and K components instead of having the commas in there. And so we can, we already have that position vector. So let's lean towards finding the linear momentum vector and using the cross product between the two to calculate our angular momentum. OK. So we think about the linear momentum P all this is gonna be equal to the mass M multiplied by the velocity V. Now again, we're given this velocity vector. And so that makes this much simpler. So what we have is our mass 5.2 kg multiplied by our velocity vector. Again, putting this in terms of the I hat J hat and K hat components, we can write it as negative or I hat plus 3.2 J hat minus 3.8 K hat. And that is in meters per second. OK. So we have our R vector, we have our P vector. What we're left with doing is taking the cross product. And now before we take the cross product, let's actually simplify our P vector. So our linear momentum vector, we're just gonna distribute our 5.2 kg to each term to each component of our vector. Remember when we're multiplying by a scalar, it just goes in and multiplies to every component of our vector. So we have negative 20.8 I hat plus 16.64 J hat minus 19.76 K hat. And now our units are in kilogram meters per second, maybe multiplied kilograms by meters per second. OK. So now that we've simplified that we're gonna go ahead and calculate this angular momentum. OK? And again, the angular momentum is gonna be the cross product between these two vectors. And so L is going to be equal to three I hat plus four J hat plus five K hat meters crossed with negative 20.8 I had 16.64 J hat minus 19.76 K hat kilogram meters per second. Now, we can see we have meters for the units in the first term, we have kilogram meters per second for the units in that second vector. So when we do the cross product, we're multiplying corresponding terms. OK? So we're gonna get kilogram meter squared per second for our units, which is what we had in the answer choices. So our units up to this point match up. Now we're gonna go ahead and do the cross product. There's a couple of different ways to go about doing the cross product. So whatever um your professor has taught you or whatever, you find the most effective way for you to do the cross park product, you can go ahead and do that. What we're gonna do is we're gonna say that we know that the cross product between these two vectors is going to be equal to what? Well, it's gonna be the Y component of the position vector ry multiplied by the Z component of the momentum vector P minus the Z component of the position vector R multiplied by the Y component of the momentum vector P. OK. That's the IH A component. So Rypz minus Rzpy. Now we're gonna keep going and we're gonna use the same notation. So if I say R I'm talking about the position vector P, we're talking about the angular momentum vector and then XY or Z just indicates which component. OK. So what we're gonna have next is we're gonna have RZ PX minus RXPZ. That's gonna be our J hat component. And then we're gonna add, we have RXPY minus RYPX and that is our K hat component. OK. So now what we need to do is fill in all of these values. OK. So we're starting with the first one, we have the Y component of R OK. So the J hat component that's 4 m multiplied by the Z component of vector P or the K hat component negative 19.76 kilogram meters per second. And we're subtracted K hat or Z component of R 5 m multiplied by Y or J hat component of P 16.64 kg meters per second. And this gives us our I hat component. We're gonna do the same for the J hat component. So what we get is we get 5 m multiplied by negative 20.8 kg meters per second minus 3 m multiplied by negative 19.76 kg meters per second. This is our J hat component and then we have to add our K hat component. We're gonna add it down below because we're out of space. And it's the exact same process. We get 3 m multiplied by 16.64 kg meters per second minus 4 m multiplied by negative 20 0.8 kilogram meters per second. And this is again our K hat component. OK. So now we have our L vector written out and we've substituted in all our values. It's all written out. The only thing we have to do is simplify. So we're gonna go through, we can use our calculator. We're gonna simplify each component. So four multiplied by negative 19.76 minus five, multiplied by 16.64 for that I hack component and so on and so forth. OK. And when we do that, what we find is that this vector L that angular momentum vector is gonna be equal to negative 162.24 I hat minus 44.72 J hat plus 133.12 they have. Ok? And again, this is in kilogram meter squared per second. Ok? So that is the final answer we were looking for that is the angular momentum with respect to the origin of this block. The last thing we need to do is round and in the answer choices, they rounded to two significant digits. So if we round to two significant digits, a negative 162.24 while with two significant digits that's closest to negative 160. So we have negative 160 I ha negative 44.72. That's gonna become negative 45 J hat and 133.12 that becomes 130 when we're just using two significant digits. Ok? So 130 K hat and we keep that same unit kilogram meter square per second and that is going to be our final answer for this problem. Let's go up and take a look at those answer traces again. Now what we can see is that the correct answer is going to be option D. Thanks everyone for watching. I hope this video helped see you in the next one.

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(II) A particle is at the position (x, y, z) = (1.0, 2.0, 3.0)m. It is traveling with a vector velocity (-5.0 ,+ 2.8, -3.1)m/s . Its mass is 4.3 kg. What is its vector angular momentum about the origin?

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