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Problem 70 Hard Difficulty

The position of a particle with mass $m$ traveling on a helical path (see Fig. 45$)$ is given by
$\vec{\mathbf{r}}=R \cos \left(\frac{2 \pi z}{d}\right) \hat{\mathbf{i}}+R \sin \left(\frac{2 \pi z}{d}\right) \hat{\mathbf{j}}+z \hat{\mathbf{k}}$
where $R$ and $d$ are the radius and pitch of the helix, respectively, and $z$ has time dependence $z=v_{z} t$ where $v_{z}$ is the (constant) component of velocity in the $z$ direction. Determine the time-dependent angular momentum \overline{L} ~ o f ~ t h e ~ particle about the origin.

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Physics 101 Mechanics

Physics for Scientists and Engineers with Modern Physics

Chapter 11

Angular Momentum; General Rotation

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Video Transcript

to find the angle of momentum. We're going to use a question 11.6 with, say's that angle. A momentum can be written us the prospect act between position, vector and moment Victor. So let's find the petition. Victor forced still Arkan region as radius our time scores to buy zing over big which is the Tita. So this is the X component. Similarly, we have signed by Z or will be CK cap Now see here can recognize Visi Dynasty and therefore Vesey. Yeah, since he can return as easy time Steve So in either. So wherever we have c began used this expression off the over there so basically easy over here, here and here can be replaced by the easy times Steep. Now let's find the velocity vector. So this is basically the differentiation off the position vector with respect to time. So, doing that, do you get negative? Odd. Don't buy Z. It is actually Visi time sti and then you have to differentiate that with time so ultimately arrived at this expression. So notice that you can only do the differentiation after using this expression off, see in art and then differentiating r with respect to time to get the velocity. So this is the Michel Fournier and then, yeah, the music kick up Now since the Taj Oh, there are a lot off mutation those So to simplify them, it's used all fart will be equal to do. Bye, Visi over, baby. So doing that can write velocity vector to be equal to negative Al Faw Woz sign. I ve I t I get it Blessed I for large course J cap. Bless. Course I 30 Jacobs Less easy key cap. Similarly, for the position Victor, we have cause I party. I get it. Place, Sarge. Silence. Might be dick. Yep. Yes, Vesey, Get get. You can also use this expression off Alpha beforehand as in before doing this differentiation. So basically, this will be a first step. This will be the second step. And this will be your thoughts. Stick on. After that, you can use the differences. And after that, you can find the velocity by different shipping. Are over tea on DDE. You'll get this as your fourth step. So this might save a little bit off a four decked. You have to You might have to do if you go the long way and use this expression later. Yeah, so no, we can find angular momentum using by doing the cross product on form. Oh Mental. We know that moment of majestic Aleman Stang's velocity So the news that on for the cross president really use determinant drool So doing that you arrive at that Let me just write though Tell me Knight So we have lost I can't do you get a giggle, Tell the unit directors then we have to like the confidence off AJ the night last We have the right the components off velocity vectors and then we have to take the did eminent off the smack tricks. So after taking the determinant on DDE, simplifying the expressions bit so I'm skipping the small details off Sipowicz simplifying the expression taking the determinant. I assume that you know them already. If you do, then you can follow the textbook on their 20 off examples on them. So I think the out of space. But let me just try to fit on their domes. This is still jade going Woman moves the vehicle. Let me move this as well. Oh, gonna be okay. So that's right Dad. The last we have to buy AJ oh, would be keep up. And this is the final expression for the Angela Millington. Victor. Thank you.

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Problem

A boy rolls a tire along a straight level street.…

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Physics for Scientists and Engineers with Modern Physics

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Watch More Solved Questions in Chapter 11

Video Transcript

Douglas C. Giancoli

Physics for Scientists and Engineers with Modern Physics

Marshall Styczinski

Zachary Mitchell

Aspen Fenzl

Meghan Miholics

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Physics for Scientists and Engineers with Modern Physics

Marshall Styczinski

Zachary Mitchell

Aspen Fenzl

Meghan Miholics

Douglas C. GiancoliPhysics for Scientists and Engineers with Modern Physics

Marshall Styczinski

Zachary Mitchell

Aspen Fenzl

Meghan Miholics

Douglas C. Giancoli

Physics for Scientists and Engineers with Modern Physics