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Problem 84 Hard Difficulty

(III) Figure 39 shows a thin rod of mass $M$ and length $\ell$
resting on a frictionless table. The rod is struck at a distance $x$
from its $\mathrm{CM}$ by a clay ball of mass $m$ moving at speed $v$ . The
ball sticks to the rod. (a) Determine a formula for the rotational motion of the system after the collision. (b) Graph the rotational motion of the system as a function of $x,$ from $x=0$ to $x=\ell / 2,$ with values of $M=450 \mathrm{g}, m=15 \mathrm{g}$ $\ell=1.20 \mathrm{m}, \quad$ and $v=12 \mathrm{m} / \mathrm{s}$ . (c) Does the translational motion depend on $x ?$ Explain.

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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

no since I am 11 went on for the whole system, which means the tape less rod system is conserved. We can write alive Toby Quinto left now And I is since the Lord is addressed, Ally is the angular momentum for the play with respect to the center off months off the road. So we have mass times. Velocity off quickly times exclaimed on its sea or Exley is the distance off the plea from the center off Mask off the whole system So that is a quinto. Now, since they both stick together and the Lord on dhe, the road starts rolling. Therefore both the road and typically have the same final and level Osti. So we get omega after the equal door and the extreme for But I this I see now we need to find ex C i. R. And I see. So let's first find XY first XY, as they told, is the position off the plea with respectable the sink that off my ass off the whole system. So for that, we need to find the center off mas off the whole system, so this will be equal. Do m Times X, which is where exist the distance off the stand after play from the center of mass off the rod, less capital and which is Masa? The raj hangs the distance from the center of the Lord to the center off miles off the road. And that distance is basically zero because they both are at the same point. This gives ex off center off my ass off the whole system. Toby Small in next. Oh, the turbine loss. Now we can find XY so x, See, if you remember our relating velocities abstraction. So exhibit is position off, Steve off the play with respect to cml off the respect. Oh, the center off months of the whole system will be equal. Do so Let me like dad over here. So this will be a cold. Oh, tradition off. Let me with respect. Oh, the center of mass of the Lord plus distance The going off position off center off months off. The Lord with respect. Oh, the center of mass of the whole system. So this is basically little velocity addition. Not since the center off my ass off the Lord is at the center of Mass at is at the center off the road on dictating this Toby the origin. Therefore, this distance is physically X sending off moss negatives in that off must, because this is a center of mass of the Lord with respect. Oh, sent it off. Lost off the system. And so this is basically position off the origin with the respect of the center of mass of the system, which is negative off the position off the center of mass of persistent with her Specter order agent. So there will be a ticking negative site so exciting it before all of this exhibition on be equal to X, which is this value provisional fix. With this descended on Mars is the god Linus X, the M s. So using this expression now over, he'll get lexie to be equal to capital and bangs X well, but And bless capitalism. No, we have one of the really believes that we need the substitute here. Then we need I. I see. Let's write, I suppose because ah, it's a little bit simpler now after the Lord after decay has stick, tow the road So one of the ends of the rod, it starts moving around the center off the road on. We can treat the play to be a point must therefore the moment of initials. The play will be Marcel tickly. It is small, um, fines this land square. We have excess. Remember that now we're solving everything with respect. Oh, the center of mass off the whole system. So instead, off x, they took exceeds now for the Jord. So if we had steak way had, uh, bean solving everything with respect to center off months off the road Only then I added the speak will tow center of mass of God. The Lord is you're taking about the center. So this will be the case. This will be the value. But here we are finding moment of inertia with respect to the center off north of the whole system. So, basically, CMS, Because of that, we need to use battle Alexis tomorrow. So we have to add another time, which will be mass off the road times the distance from the sink that off the Lord do the center of mass off the Lord and the center off. The George is basically the artisan. So this would be 1/4 then Capital and Mills grab the capital. Um, so just Cecile and this is basically necks See, in this let's be found here. So we use that over here and we get now they can use I r i c on XY. In this equation, defined only got f So maybe I have with the equally true and the XY overs. I, uh bless. I see. And if your substitute each of these values this is equal to a and B and mix. This is ai R plus our signatures and excess girl. But again, in place off exit, we have to use the expression what exceed that we found before. Now, if you simplify this a bit, you see that a lot of these stones cancel out and finally a rise, Adam, make I have to be equal to three times x over. I know what you is times one plus capital and over small m and scratch. Yes, exclaim. So this is the final answer off. Final expression for omega. If now you guys think so. It's Graf dysfunction omega f the respect to eggs on. Let's take the range off X from nine. Do sit a point six meters. So for us Let's like this expression again. So we have omega after the equal do V X over one over one less capital M. Well, what am I? It's good this X squared Now. This can be written for the values given floor three Capital, um, and smaller. And this can Village in Toby called 12 x over 2.72 plus X squared the young itself. The unit over Megan is readying for a second. No, with the plot omega after their spectral x uh, taking the range of extra B cups. So let's call this. Let's say we have taken during from 0 to 0.6 meters. So if you do that so my graph won't be exactly to scale, but you'll get it all fighting off how the graphs should look like so they have omega in the Y axis, X and X axis on. If you brought, let's say you did go six points. 46 points will be enough. Put this explains for X, ranging from 0 to 0.6. So at zero we have Omega Tau Basile. Then at that 0.2, omega will be some veg around zero point seven. It's not this here on Dhe. Similarly, you can mark other points and it said 0.4 on *** is one point to fight. So if you keep on doing that, you'll see that after the vial of Omega starts bending towards XX is so this Will, this girl will not be Ah, complete straight line. But on the slope will be closing off to 45 degree. But it won't be a straight line or you don't be completely near function because of this extra time over here. But gradually Omega will start. Uh, ending start up. Getting close towards XX is now the spaceship used for this problem can be found on the medium and file B s, C four, Chapter 11 Extension of the violence Excel s on DDE. This is on stab, but all of them 11 point 84 b now, last but not least, we have to do part. See, it is quite simple. So Alina momento is guns off in a totally in a complete in elastic collision. So even though kinetic energy is not conserved But Lena momentum is always conserved for a completely interested conditions. So we have bee. I called B F momentum is basically mass times velocity. So we have n times the equal toe. So for the final motion, we have both. The moss is having the same final velocity because the stick together, that's why the motion is the collision is called completely inelastic collision because these two months is stick together. So from here we can find the until the equal do small Emmy. I know all the good in us so from this expression. So this velocity final velocity is with respect to the center of mass. So from this expression, we can say that the translation mull motion or the velocity of the center off Mars is not dependent on X because there is no X in the secretion. Therefore, our translation motion is not living in our necks. Exes Only X only comes into a conch for a rotational motion off the body.

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Video Transcript

Douglas C. Giancoli

Physics for Scientists and Engineers with Modern Physics

Elyse Gonzalez

Christina Krawiec

Aspen Fenzl

Jared Enns

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

Physics for Scientists and Engineers with Modern Physics

Elyse Gonzalez

Christina Krawiec

Aspen Fenzl

Jared Enns

Douglas C. GiancoliPhysics for Scientists and Engineers with Modern Physics

Elyse Gonzalez

Christina Krawiec

Aspen Fenzl

Jared Enns

Douglas C. Giancoli

Physics for Scientists and Engineers with Modern Physics