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Question 35 ptsA spherical bowling ball rolls without slipping along a flat; horizontal wood surface with constant velocity of 2 m/s. It then comes across a patch o...

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Question 35 ptsA spherical bowling ball rolls without slipping along a flat; horizontal wood surface with constant velocity of 2 m/s. It then comes across a patch of ice (with NO friction between the ball and the ice). What happens to the ball?It stops rotating, and slides with V > 2 m/s:It stops rotating, and slides with v = 2 m/s. stops rotating; and slides with v < 2 m/sNone of theseIt continues to roll; but now with v > 2 m/s It continues to roll with v = 2 m/s.

Question 3 5 pts A spherical bowling ball rolls without slipping along a flat; horizontal wood surface with constant velocity of 2 m/s. It then comes across a patch of ice (with NO friction between the ball and the ice). What happens to the ball? It stops rotating, and slides with V > 2 m/s: It stops rotating, and slides with v = 2 m/s. stops rotating; and slides with v < 2 m/s None of these It continues to roll; but now with v > 2 m/s It continues to roll with v = 2 m/s.



Answers

A tiny solid ball $\left(I=2 M r^{2} / 5\right)$ rolls without slipping on the inside surface of a hemisphere as shown in Fig. $10-12$. (The ball is much smaller than shown.) If the ball is released at $A$, how fast is it moving as it passes ( $a$ ) point- $B$, and $(b)$ point-C? Ignore friction losses. [Hint: Study the two previous questions. When it comes to the ball's descent, its own radius is negligible.]

In this problem, we need the soul for the linear and angular velocities of each ball after the impact, for better and for letter me, the velocity off ball be after it has started rolling uniformly, so the impact is assumed to be perfectly elastic. So while a with Mass M and readers are rolling without slipping with velocity V O at an angle data and angular velocity omega oh, strikes and identical Boiled B, which is initially at yes. So it is the point of the body, um, hit will be Is that point being so this produces an impulse and causes ball A and will be the role at angular velocity. Omega and Omega be, respectively, and for their center of mass is to move with velocities. B a n b be respected. So the answer Never E. Let's look at the impact on both balls. So I have here a sketch for ball um e so you can see the velocities, then the direction of developed velocities as well as the angular linear velocities and angular velocities. Okay, so since ball is initially rolling without sleeping so we can see that so omega sub o equals V over R Now, applying the principle of impulse and momentum and writing the equations for the linear components and for the moments about X and y axis will have the follow being equity Shawn's. So for the horizontal in your components. So we have So am the old go saying lida was a CNC are we owe here is directed at an angle, then minus the impulse B B P equals So am B A X So this is our agree usual on one. Next for the pretty girl being your components. So we'll have to m b o sign data blood zero equals. So am V a white. This is our equation. Do and then so living for the moment about the y axis. So we'll have the ratio. And I Omega o course I did zero because I omega a cause. I be that so you can see this, um, the last part off the sketch. So we have to be there this Air equation three and then lastly for the moments about the X access so it haves and negative I Omega o science data plus zero equals negative. I omega. A sign beta. So this is our agree for four. They're So this is for, um, boiled a now for, um, ball be so we'll write the same set off equations so we'll have the four years on. Does leaner components will have zero plus the impulse BDD equals o m times be x. So this is our equation. Fight and then diverted guy. So for the read the gun in your components, so have zero plus zero equals m. B. Why? So this is our equation six and then serving for the moments about white ass is so we'll have again. Zero plus zero equals. I will make me co sign Gamma. So again, this river do the sketch for this one, so you can see that we have angle gamma there. And then, lastly, moments about X axis Leisure one. So with abs, 00 equals. I got me sign Gamma. So this is our agree show on eight now. So adding equation one and five will have I think Equation one and I will have, uh, am really old Gaza. Indeed. A equals am the X plus M v b x. So this can me I am can be cancelled out from both sides of this will become the old cause. I'm data waas the a X plus B X. So this is our equation name and then rating another equation from the equation for the coefficient of restitution with e equals to one, since the impact is perfectly elastic. So we have So e is able to negative the a X my nose vb X over Rio or science data my nose zero. So sens e is equal to one began, right this us, um, the old course and data equals. So we be ext my nose Me a X. So this is now our agree Sean, then now so so being four Equation nine and they and night and day and simultaneously. So when have or will get BB eggs equals the Opus ndta v A X will be initially cancer. Then once you saw the X, so you will see that this is equal to zero and then from immigration to and six will get from vacation to end six will get so v a y because so we owe Sign that and be wise. It will do zero hand so we can know get being a and B B so we'll see that b a is equal, Do we? Old science data So which has only wagon bone injuries critical and then be is equal to the O co sign data. I had two, which means it's only along the horizontal. So it's it's you really hard is on that No for so solving for equation three and four. So having for equation three and four simultaneously So we'll see that You see that beta physical data and so omega a physical door Omega o. So which is equal to be, um so Rio over art. Hence, um, Omega is equal door Be over our times Negative sign that I had plus go sighing Data j And then lastly are solving for equations have been in eight simultaneously We'll get in a time of being you will get or you will see that omega me is a good tool. Zero These are now the velocities, the velocities and angular velocities off Oh boy A and Ball B. Now for letter B for letter me. Well, look onto the subsequent motion off will be. So I have you a sketch off that particular scenario. So since the bulls rose uniformly after the impact. So this means that we be prime physical to our Omega. Be Brian now picking the moments about the incident The news point off sedation is against C in the figure. So would have and b b r plus zero equals I Omega be prime plus m b bribe art. So these in the shape off the boy it's this spear. So its moment of inertia is given by So I waas to over 5 a.m. r squared. So substituting this and the equation for a TV prime a year so this equation will become so m b b r equals so to over 5 a.m. r squared omega be brian plus m r squared omega be brain. So simplifying the equation So this will be m v b r equals 7/5 m r squared omega be bribe. So we'll get so maybe privacy too. Five vb over seven or or five. The old because indeed the over seven art hands we be bride is equal to so vb premises are times omega be prime. So so be equal toe five. I was video go sign data over seven. There's a real force indeed I had, which means that it's along the ex office. Our heart is on

In this problem we need to solve for the linear and angular velocities off each sphere after the in back. So that's where they ate and for leather me the velocity off each sphere after they have started rolling uniformity. So the Outback is assumed to be perfectly elastic. So Sphere a off Mass, AM and Regis are, which is a ruling without slipping with Velocity V one and Angular Velocity Omega one hits on identical Sphere Me, which is initially a crest. So this produces an impulse and process. Sphere A and sphere be the role with angular velocity, Omega and Omega be, respectively, and the movie Velocity V A and B, respectively. So let's first look into the in back on spirit eight. So I have year a sketch off the, um, scenario. So for sphere A. So since, according to the problem, Sphere is initially rolling without slipping so we can see that omega one Seiko toe be won over our now applying the principle off, impose and momentum and taking the moments about pointy, which is the center of this fear, with the clockwise direction, and begin as the positive direction. So we'll have. So I Omega one plus zero equals. So I Amanda eight. So this, um, from this, we can see that Omega one is actually equal Do omega A. So which is God through. So this means that omega is also equal to be one over r No writing an equation showing for the linear components off momentum over again for Spirit e were still on spirit. So we have. So am me one my nose. So the impulse equals am me eight. So this is a word equation one. Now let's look in tow. Impact onto the impact on this Fear me this time. So I have your A sketch for spirit. Be so again, applying the principle off impulse and momentum and writing an equation for the senior components momentum. So when have so zero plus with the MBA's B B B equals I am we mean So this is now our equation on do now I think equations one into so the impulse would be cancelled out. So we'll have so end. The one equals O M B A plus M V V. So the masses would actually cancel out. So this will become so be one equals B a plus B B we can have. This is our equation now, writing another equation from the equation for the coefficient of restitution or so we'll have the equation for the coefficient of restitution is equal to so easy Will do negative. So be a my nose baby over. We want my nose zero. So it's gonna be then S o E v one. It waas maybe my nose we ate and since is equal to one because of the impasse Perfectly elastic. So we have so be one equals we be my nose the A So this is our equation for now Taking the moment about point eat again the center off this fear with clockwise direction taken as positive So we have, um, zero zero equals So I will make a b Iomega Be So this gives us Omega B is equal to zero now solving for equation slee and four simultaneously so we'll get it's right that that's all being poor, racial in sri and equation for until, um B a would be cancelled out so will be left with so v b equals B one. So solving for the eight. So you'll see that b A is equal to zero. So hands our answer. Where letter is so the velocity of this fear is good. Zero spear a zero and then it's angular. Velocity is equal Doe V one over R in the clockwise direction and then for spear me So its velocity. Zico Group B one And then it's angular. Velocity is equal to zero. Sorry, so VB is to the right. Then angular velocity is zero. Now let's look onto each arm spheres motion after the impact. So for us fear a So for sphere A. Since this fear rolls without slipping, so would have the A. Brian is a photo omega prime times very juice. So next solvent for the moments about the instantaneous point off rotation since its again rolling without sleeping. So we have, um, I Omega E Plus zero equals Iomega a prime plus M v. A prime times are so given the shape off this fear. So its moment of inertia is given by door over five m r squared. So substituting this and equation for me um, A and Omega A from part A. So are a creation would become so too fibs m r squared times the value of Omega averages be won over or two equals 50 m r squared omega A squared loss So M r squared Omega e squared. So simplifying this will have So do it's m or anyone equals 7/5 m r squared Omega e Brian. So from here we can get Omega A. So make a Prime is equal to do V one over seven Art. And since we a single omega, a prime times are so be a primacy Go Duke to be won over seven. Next, we'll write the same set of equations. Four. Spear being so bored Sphere be first, Fear bees again, since it's also rolling without sleeping. So we be prime. Itwas omega be prime times are and then for the the equation for the moments about the incident in this point off sedation. So we'll have. So am M v. Me Times or blood zero equals I omega Be prime plus M v b Prime times or so something something decoration for, um vb Brehm and I so will have m v one r equals over five m r squared omega be prime plus m r squared omega me, Brian. So this would be UM m v. One or because having over five m r squared omega be brain. So from here can get Omega Be prime equals So five b one uber seven R and then scenes. The B prime is a gold toe omega Be Brian times ours a baby prime musical toe. Five Be won over seven hands. The answer or let me is so v a Brian is a go to do be won over seven and then b B Prime is equal. Do five B one times be won over seven.

In this problem. We have four small balls, each with identical mass N b a moving with velocity V towards C. They strike C and C then strikes d. Now, if we know the final velocity of CND which is V you want to know why d will not move off with velocity to be so? The collisions will occur in the following following sequence. Firstly, ball be strikes ball see and we'll take motion to the right as positive. So using the conservation of momentum, we see that M v which is d momentum for ball be remember ballsy stationary So its momentum is zero is equal to minus AM vb the velocity of the after it strikes. See bless em V c. So since the masses identical, we can divide both sides by m and we get V is equal to minus VB plus V c. Now we could also use the coefficient of restitution here and take again. The right direction is positive. So e is equal to one the coefficient institution and this is equal to V C plus vb over the initial velocity V. So essentially we have two equations with both unknowns. VV NBC if we solve them. We find that we see after the collision is V and VB is equal to zero. So essentially the B comes to rest and we see now moves off would speed the The second collision is ballsy striking for deep and again, we use the methods above. Taking the right is positive and using the conservation of momentum first, so see moves off initially with momentum M V and after it collided with Dee Dee. Momentum is due to C minus M v C plus m the d. So we can write this if divide both sides by m v is equal to minus V. C Bless VD. Now we do the same as it above. We look at the coefficient of restitution again. That's one So e is equal to one which is V D plus V C off the by definition, and so we can see from here. That we see after the collision is now zero, and VD is equal to be now. The next collision is bought a striking Barbie using the conservation off mentum. We have a moving at initial velocity M V, and this was equal to minus um V a l s m vb So again we could divide through by m and we get the is equal to minus V a plus vb so we can see a pattern emerging again. The coefficient of restitution e is equal to one and this is VB plus V a All over v. And so again, two equations with two unknowns gives us that VB is equal to me and va comes to rest. So the A is equal to zero and for the final collision, Paul be then strikes bold seat. Now again, we'll use the conservation of momentum and M v. The initial momentum of all be when it strikes policy again is minus m vb last the momentum of ballsy M v c So we is equal to minus b b les d c. And again if we apply the horizontal components of the coefficient of restitution yeah, is equal to one and it's defined as V c yes, VB over the And so again we're left with two equations with two unknowns. We get V C is equal to V and Bobby comes to rest. So VB is equal to zero Now what we can see from here is that if the rolled off with twice the velocity to V, it's kind of like energy will be twice the available from the original two A and B so half and V squared, that's a half, and the squid is not equal to my house and to the or square, so this is not possible.

So this question has two statements and a statement. one. It is saying that if it's peer rolling on a horizontal surface has elastic collision normally against the smooth wall, then it's written motion. Then it's written motion will also be a pure rolling. Okay, you're rolling. So this statement is incorrect statement because the pure rolling motion will not be occurred after collision. Okay, because the angular velocity do not change, but the direction of velocity will get rivers. Okay, this will get rivers, so hence we can say that the rivers motion will not be a pure rolling motion, it will start sleeping, okay, it will start sleeping. So this statement is wrong statement. And moving to the statement secondly, says that its speed and angular velocity do not change after such elastic realism. Okay, there is elastic realism. So hence we can say that speed and angular speed do not change. So this statement is correct statement Okay, directions will be reversed, but magnitude will remain same. Okay, so this statement is correct. So hence we can say that option D is correct answer. Okay. Option D will be the correct answer for this question.


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