5

Find the indicated moment of inertia or radius of gyration.A top in the shape of an inverted right circular cone has a base radius $r$ (at the top), height $h$, and...

Question

Find the indicated moment of inertia or radius of gyration.A top in the shape of an inverted right circular cone has a base radius $r$ (at the top), height $h$, and mass $m$. Find the moment of inertia of the cone with respect to its axis (the height) in terms of its mass and radius. See Fig. 26.57

Find the indicated moment of inertia or radius of gyration. A top in the shape of an inverted right circular cone has a base radius $r$ (at the top), height $h$, and mass $m$. Find the moment of inertia of the cone with respect to its axis (the height) in terms of its mass and radius. See Fig. 26.57



Answers

Use cylindrical or spherical coordinates, whichever seems more appropriate.

A solid cylinder with constant density has base radius $ a $ and height $ h $.
(a) Find the moment of inertia of the cylinder about its axis.
(b) Find the moment of inertia of the cylinder about a diameter of its base.

Yeah. Were given a solid and rescue cylinder towards your coordinates to find in part a the momentum of inertia of the cone about its axis. Syria, yes. Yeah. So the solid is a right circular cone with a constant density for the beast radius A and the height. H never heard this on. So I guess the question first is what's the best way to describe this salt, cylindrical or spherical coordinates? I would say. I've never seen. Maybe some wonderful coordinates is easiest. It's not clear how this relates to a spear exactly now. Yes. I'll call the density delta. Okay. Yeah. Right. And we'll put the vertex of the cone at the ordinance and we will face upwards. Yeah. So you're not we project the cone on the xy plane. We have data will lie between zero and 2 pi And the Radius R. lies between zero and the radius of the basic income. Which is a and finally the height. Uh huh. Well this is gonna lie between effectively 50 has to be greater than or equal to the square root of X squared plus Y squared. Which in this case is square of R squared. Or just are. And the height of the cone is H. So Zs be less than equal to H. EMC you all like. No there right like two. There is three brothers Now to find the moment of inertia of the coordinates. Access. Well from our definitions, the axis is the Z axis start finding the moment of inertia about the Z. Axis I subsea This is the angle from 0 to 2 pi in a growing from zero to A. And they grow from R two H. Of X squared plus Y squared. Which in cylindrical coordinates becomes r squared times the density which is pay times didn't which becomes our dizzy gr with data ST Now I can write this as a product of genitals and a girl from 0 to 2 pi. The data and it goes from 0 to a uh R squared. We are times the integral from Treasury are cute. Beyond And a girl from R. two H. Actually we can't quite do that. You do have an R. And the limited the integral. So we can't read it as a product for taking the derivative with respect to Z. First pay times integral from zero to pi integral from zero to A. And our cube farms Z from Z equals R. Two H. Pr the data. Uh huh. It's pay time is integral from 0 to 2 pi. And the girl from 0 to a. Of our huge times H- Art of the 4th E. R. D. P. To Taking anti derivatives with respect to R. K. times integral from 0 to Pi. That's actually this this angle from 0 to 5 ft at times in the world from 0 to a. Of our cute H minus are the fourth beyond now this is K times data And say that equals 0-2 pi times Part of the 4th. H over four minus art of the fifth over five articles. Due to a this is K times two pi block times. This is eight of the fourth. H over four minus 85th over five. Mhm. Uh huh. Right episode special guest and and a fighter. What? Yeah. Okay. Yeah. So I made a steak earlier actually the description of a cone is of these actually going to be greater than report to each over a times are two each. In this way we see that we ate sugar agents are this is a cone which has spare text of the origin and then when our reaches the radius of age it gets to a maximum height. H just makes sense. So now this is Kenny girl from Page over eight times our speech. This is equals Hard Times October 8 2 H. Which is yeah our cube plans H minus are the fourth. H over A. Mrs K. Times are cute. H minus. Yeah. So Age out of the 4th over a. Which is part of the 4th. H. over 4- part of the 5th. Each over five A. mrs Caters to High tons 84. H. R. four H. Eight of the fifth over five A. Which is H. Eight of the fourth Over five. This is the same as factor out A. The fourth. H. We have hi H shoot in the fourth. Yeah. Over let's see this is 5 24, 21 21 31:10 Over 10. And there's AK. So Kate i. h. eight of the 4th. Over time will be. Mhm. Now this is one approach to finding its moment of inertia. Can we write while leave in this form? Actually see if you can write this in terms of the volume of the code. That's my advice. Then in part B we're asked to find a moment of inertia of the cone about a diameter of its space. Well this is a little bit different. Imagine that we flip the cone So that data still ranges from 0 to 2 pi articles zero up to a. And well, let's see now range from zero up to hey. And now we have a cone with vertex in the plains equals age. It's gonna be a church minus H over a. Are. So you see that when R0 this is a church minus H. over a When are is up to a maximum radius Z is equal to zero. See Yeah. Tell me you said lisa this Yeah, there. Right. Yeah. So the moment of inertia, the climate's diameter on this case the diameter of its base is the X axis for example. Really any line to the origin in the xy plane you want to find I X. Well this is the integral from 0 to 2 pi You go from 0 to a Any growth from 0 to H -H over AR. Um Why sweats has C square between cylindrical coordinates is r squared sine squared data plus Z squared times the density which he said was. Kay, this constant times V v becomes our DZ drd data taking into derivatives with respect to Z. This is or K times in Iraq from Syria to pon. and he grew up from 0 to a uh are huge Sine squared data. Z plus our Z cubed over three. Hi Z equals 02 H minus H over A. R. He already data what? Yeah because he had the first just K. Times. The Girl from 0 to Pi and the girl from 0 to a okay his H minus feature for a R cube. No, my mistake replacing Z. Not this but this is R cubed sine squared data and H minus H over A R. Plus our over three times H minus H over A R. You let's see he already be there Russia. Fresh out of the water This 3 2nd takeoff. Now if you take anti derivatives this is K times smell it. H over 40. So that's how as if you guys are where it was just on a pussy director date of the ford. Is that what you're saying? Data journey -H over 80s. Yes, as stated ST sign of tooth data Plus HQ moved over 60. Yeah. Mhm. A square data. Listen, I'm just as simple where theater can range from zero to pi see mhm toward the end yet. Hey, times fuck. All right. Pi H over 20 maybe the fourth plus high HQ mm over 30 is swear. Mhm. We've all seen. Alright. Client knows that. Yeah, this is actually a Yeah. Mhm. Yeah. My advice to simplify try and right this moment of inertia has the sums of an expression involving the mass of the original tone as well as an expression involving the massive original cone and the height. In other words, hard to write this so that we can get rid of this constant terms case.

So we need to find the the moment of inertia of this cone so we can just draw the cone. Here we have the radius from the center of the cone to the edge. Well, actually, label this this capital r, and then we have this access going through. Put it here. And then we have this fitness associated with this cone and this thick. This changes as you go up the cone. So we can say that this is the X we're gonna see This angle here is going to be considered Alfa. And then we have the height of this entire cone. This is going to be considered a tch height. And this Vertex here, we're going to label this very tex, eh? As you go down the cone as you go down the cone, the radius changes, so we're gonna have to model it with radius are lower case R. And at this point, we can tackle the question. So we have the mass of this cone is going to be equal to the, uh that's the cone Times. The volume of the cone and the volume of the cone is going to be pi r squared h over three. So we can say that the mass ah times equals equals the density times the volume. This is the definition for mass and this would be equal to Ty row R squared H over three. Given that the volume of a cone is again pi r squared h over three. So in this case, we would have to solve for row. So Roe is going to be three em over pi r squared age. So in this case, we can say OK, let's get a new workbook and say, Let's consider one disc of this cone of fitness DX, which we have labeled here in the diagram. And then we'LL say at a distance away from Vertex, eh? And at this point, we can say Okay, our is going to be equal to X ten Alfa. So this is where it gets a bit complicated in the sense that we need to find the volume of this piece. So the volume of this peace would be, of course, pi R squared times the thickness so dx and then we can salt. We can plug the end for our and say that this is gonna be pie X squared ten squared of Alfa d X. At this point, we can say that M is going to be equal. The mass of this cone is going to be equal to pi A sorry ro times pi X squared tangent at ten and squared of Alfa Times again dx I The moment of inertia of this cone is going to be the mass times the radius squared, divided by two. And at this point, we can say Okay, if this is if this is true, we can plug in for our mass and we have pie rather row pi X squared attention squared of Alfa the Ex all over two times x ten of Alfa squared That would be our radius term. And at this point, we can say that I was in an equal who row pie of x to the fourth, ten to the fourth Alfa D X all over too. And at this point we can melt, we can integrate so we can say that I was going to be equal two zero toe h of of Rho pi ten to the fourth Alfa all over, too Times X to the fourth DX. Now this is on ly because we have DX here. So in order to find the true, um, moment of inertia, if we were to integrate this from zero to the full height of the cone, we can essentially find the moment of inertia with respect to X, where X is again that thickness. So we simply have to go for through the full length integrate folks, integrate the full length of the cone with respect to X, and we would find the moment of inertia. So this is going to be row pie ten to the fourth Alfa invited by two of age to the fifth over five, and we can say I equals ropey over, wrote wrote Pi Ro. Divided by two times are to the fourth over eight to the fourth, and this has given that ten of Alfa equals R over a tch. This is from the cone. This is a given from the cone. So if we know that this is a given from the diagram of the cone, we can substitute this and say that this is going to be our to the fourth over H to the fourth and then times eight to the fifth over five and this is going to be equal to pi. Grow our to the fourth H all over ten. At this point, we know that road is going to be equal to three em over pi r squared h So at this point, we know that I is going to be equal to Pi Ro are to the fourth H over ten times and rather let's erase this grow. We're racing this road because, of course, we've been a substitute. Four row and row here is three em over. Pi r squared h this h this h cancels out this pie. This pie cancel out and we are left with three em are squared all over ten. So this would be the moment of inertia of this cone here, and that's the end of the solution. Thank you for watching

All right, It's over. The following We have a cold here. Eggs. Why? See kun white one Meridia swarm. So something like that. These where he's, uh, symmetry with respect to from c u Want to complete their moment. Ah, about, uh and you know this your axes, we'll see the sea access so we don't need to integrate it. Rygel, these vision g, we need to make the integral far squared over Birdy. What you mean necessity for you to be one s o here. Ah, well, what exists the height? That's well, you have the radius. One work are, uh, one for Z. I'm, uh, but he's he's buying you Is value here, z people to one on the radio is also one. So the relation is gonna be that No, the real used to go to see, uh, she would need to make our c people from are to one. So you do, uh, well, being true, I should be if we're gonna make, uh, respect to their eyes all the way around here to defy hold, turn to buy. Um, so that is Sarah. You Vera. Um, but it goes from zero 21 Our dean z is gonna go from are up to one. You are born. Thanks for the sea. And then we have our volume element for the moment of inertia. He have to do times r squared. That's what we have to do it. So do it Z thing Go see, is just, uh see. So that volunteer meeting, uh, one on our hoagie You want one minus r to you? That factor to have being to be from zero to buy, you're up one one minus Liar. Well, I'm sorr cube are so so these would be during general. So the into go our cube miners integral to fire to the fore. So this will be the internal, these one you saw? No, I do the fourth floor into along these anus under the four or five my fight so that between 10 the 14 my nose, my zero, my years of that cereal. So there's a view on the fourth line of work which people tool to these people to darkness. My assumption is that so by minus four over for I'm sorry. Five Monday, and, uh, well, this would be Daddy Trilby. Turn to the angel from Dubai. Well, well, this is one one over or it was five, huh? He said. I said I would be just through by Over. Or there's five of these two concerts went through there. No, If I were 10 why were two teams fight? So that's what the that should be, these moment of inertia you.

So we have. Ah, you know what Cone one that you like will find its moment of unusual bolding. So we're gonna please. We'll hide which I want to find its moment of Asia about the suction. So it seems your every plays it up off the con on both. So that, uh, you have something like these Generally diseases there. Nice, eh? On that hide their age, So But we want to make them with the veneration. Would have to doing r squared. Our daddy says so. The taxes on union you cylindrical coordinates. Um, so, uh, well, for But bones are gonna be o r r Who's, uh, b r r goes from sear off to a Did you on these for years? Of you, you are. See, you know something like these for our peoples too. Okay, You know that behind is one of the age, so that, uh oh, here divided my well, You can complete this local grease, so those laws should be changed in their age, divided by changing their remember nephew Blufgan, are you going to say you so that the World War Z both from there to there ago? There, So are you see who? I sure these, uh, are h over a to age on the volume elements are we are integrating with respect all the way around this we have through it. Ate that over the moment of inertia is, uh, these internal that enoughto plugging on r squared so that yeah, So you have. So they're insulting. Power should be, are cute. And then yeah, So the moment in your jammies Gimme that moment. All right, So, um, go ahead and do it. Seita seems nothing. Their defense, huh? So you just got to buy on dinner for being drill. We'll see. We dream. We see. So the travel logs here we see, Just see, one work is leaving. Are I'm saying you're a, you know, seething are no. That's the lower body on Love's a bridge. One of these is able to age Linus for our age or what? Um, right. So that you exist, Joe. We'll see now. Helping to make we sparred are but things are cute. So the remaining into Louisa these So each minute mar Well, e I'm sorry, Cube. We are on Duh. Not from zero up through. No way. So all these first thing tomorrow. First the jewelry arc you pidge the seven wonders minus R to the full of flour H m c So when we integrate those two, we're gonna get our cure result art of their full power. Therefore, his age minus are really each other far too. The fourth Art The fifth power by five sage for a So these volatile in zero on a news. Um hey, to the floor power. They're for him. Sage miners. A degree is powered by five. I'm sage. You're a five. So these people do these councils one hour, then you have you before age, then four. Your message Them's 1/4 minus one. If so, these have even to five brows. Motive behind that. That was not my thing. Is that so? Five minus four over all times five. So this is a vehicle. Something is able to dad and then get the factor of buys you in there so that the total moment of illusion it's quiet should be his number. That is gonna be, But he's one. We're so 1/4. Power him sage. That was one over tow. Any that is board inside. And then that and you too. Bye. That I'm still buy. When you gonna simplify that yet by a riff off inch. And this is another thing. So you should be the moment of it. The moment the range of the school.


Similar Solved Questions

5 answers
Lenriae4 Whe unund Ae ea lece4s Eael ICtn Eenutd Eent J#e noruic a Hatliaatuah Fual Fnd ofrl Tekh huillkhsssa hutha hcighi ahnt t Inc gound thc natet srile luleTSnlili[0 e Jocun eslove H hILS Ihe buildingCrcertat&
Lenriae4 Whe unund Ae ea lece4s Eael ICtn Eenutd Eent J#e noruic a Hatliaatuah Fual Fnd ofrl Tekh huillkhsssa hu tha hcighi ahnt t Inc gound thc natet srile lule TSnlili [0 e Jocun eslove H hILS Ihe building Cr certat&...
5 answers
One of the Maxwell relationships that we derived in class was:25Using this relationship; and the equation for the van der Waals equation of state; derive the correct expression for the isothermal change in the entropy of a gas (which can be described by the van der Waals equation) when you change the volume from Vi to Vz_
One of the Maxwell relationships that we derived in class was: 25 Using this relationship; and the equation for the van der Waals equation of state; derive the correct expression for the isothermal change in the entropy of a gas (which can be described by the van der Waals equation) when you change ...
4 answers
T OI nexagonal3.41 Determine in the the indices following for the directions hexagonal shown unit cells:4303(b)
T OI nexagonal 3.41 Determine in the the indices following for the directions hexagonal shown unit cells: 43 03 (b)...
5 answers
Anton Craptar 4, Scctlon 4.8, Questlon 072 Find the largest passlble value for the rank of and the smallest possible value for thc nullity of A AIs 7 * 7 The laraest possiblc value for the rank af A 15The smallest posslbla valuc for tha nullity 0f ^how HInILink T0 TtkT
Anton Craptar 4, Scctlon 4.8, Questlon 072 Find the largest passlble value for the rank of and the smallest possible value for thc nullity of A AIs 7 * 7 The laraest possiblc value for the rank af A 15 The smallest posslbla valuc for tha nullity 0f ^ how HInI Link T0 TtkT...
5 answers
Two objects collide while traveling in opposite directions. The collision is perfectly inelastic, with m = 4.0 kg, V, = 8.0i m/s, mz = 6.0 kg and Vz = - 2.0i m/sFind the velocity of the objects after the collision. What is the change in kinetic energy due to the collision?
Two objects collide while traveling in opposite directions. The collision is perfectly inelastic, with m = 4.0 kg, V, = 8.0i m/s, mz = 6.0 kg and Vz = - 2.0i m/s Find the velocity of the objects after the collision. What is the change in kinetic energy due to the collision?...
5 answers
Ba La HO Ta J Re 189 13L } 138 9 178 5 180 9 186 ! 87 SS 89 104 105 106 107 Fr Ra Ac RC Db Bh 1262 S3 (i @6 471 12611 (264158 59 60 61 62 Ce Pr M Pu Su Hal 109 4LlS J 90 91 92 9} 94 Th Pa U N Pu 320 2H10 28.0 (40A 0.2146 moles of Mg Cl20 B. 0.51006 moles of Mg Cl20 c 0.1353 moles of Mg Cl2 0 D. 0.3240 moles of Mg Cl2
Ba La HO Ta J Re 189 13L } 138 9 178 5 180 9 186 ! 87 SS 89 104 105 106 107 Fr Ra Ac RC Db Bh 1262 S3 (i @6 471 12611 (2641 58 59 60 61 62 Ce Pr M Pu Su Hal 109 4LlS J 90 91 92 9} 94 Th Pa U N Pu 320 2H10 28.0 (4 0A 0.2146 moles of Mg Cl2 0 B. 0.51006 moles of Mg Cl2 0 c 0.1353 moles of Mg Cl2 0 D. ...
5 answers
Let G be a group_ Let E and F be normal subgroups of G such that G/E and G/F are abelian. Let G/E x G/F be the group homomorphism such that O(g) (gE; gF) for each 6 G. By applying the First Isomorphism Theorem to 0. show that G/(E n F) is abelian. (No marks will be awarded for proof by a different method: ) (b) Must there exist a normal subgroup K JG with the following property: given LJG; then G/L is abelian if and only if K < L?
Let G be a group_ Let E and F be normal subgroups of G such that G/E and G/F are abelian. Let G/E x G/F be the group homomorphism such that O(g) (gE; gF) for each 6 G. By applying the First Isomorphism Theorem to 0. show that G/(E n F) is abelian. (No marks will be awarded for proof by a different...
5 answers
Let's denote the six elements of GL(2,Z2) by0A =[8 HJ s= [' %c= [? & p = [' &e= [? 4J a) what is the order of each element? (b) how many subgroups of order 2 does this group have? Please list them(c) how many subgroups of order 3 does this group have? Please list them
Let's denote the six elements of GL(2,Z2) by 0 A = [8 HJ s= [' %c= [? & p = [' &e= [? 4J a) what is the order of each element? (b) how many subgroups of order 2 does this group have? Please list them (c) how many subgroups of order 3 does this group have? Please list them...
1 answers
Convert each degree measure to radians. Leave answers as multiples of $\pi .$ $$-900^{\circ}$$
Convert each degree measure to radians. Leave answers as multiples of $\pi .$ $$-900^{\circ}$$...
5 answers
Question 16 The following diagram is of a galvanic cell: Answer the questions related to the cell.IM Pdso4IM Fcuom Imfaioy;Ppt20 $+E"= 01Jv E*=0TTVWhich of the following is the balaniced equation for the reaction taking place in the cell? Pbz- Fe* Fez b, Pb Fezt Pb?: Fe# ZpL Fez+ ZPb"* Fel d Pb 2Fe" Pbe* ZFez
Question 16 The following diagram is of a galvanic cell: Answer the questions related to the cell. IM Pdso4 IM Fcuom Imfaioy; Ppt 20 $+ E"= 01Jv E*=0TTV Which of the following is the balaniced equation for the reaction taking place in the cell? Pbz- Fe* Fez b, Pb Fezt Pb?: Fe# ZpL Fez+ ZPb&quo...
5 answers
A painted surface area of 0.75m2 on a building is heatedduring the day to 30C. Its emissivity is 0.9 and the airtemperature is 20º. The approximate amountof heat generated is ______W.
A painted surface area of 0.75m2 on a building is heated during the day to 30C. Its emissivity is 0.9 and the air temperature is 20º. The approximate amount of heat generated is ______W....
5 answers
(30 Points) Consider the function 2r2 + 82 + 8 f() 322 + 12 Find all vertical and horizontal asymptotes for f(r)_ Use limits to justify your answer . Show all of your work
(30 Points) Consider the function 2r2 + 82 + 8 f() 322 + 12 Find all vertical and horizontal asymptotes for f(r)_ Use limits to justify your answer . Show all of your work...
5 answers
Which of the following differential equations is separable:dy =sin (ty) d&d = et+y dtd = e' y dt dy =t-Y dt
Which of the following differential equations is separable: dy =sin (ty) d& d = et+y dt d = e' y dt dy =t-Y dt...
5 answers
All that is known concerning mysterious second-ordcr constant-cocfficient differential equation y" + py f(t) is that t2 + 1 + cos t, t2 + 1 + e' sint; and t2 + 1 + cos t | sin t arc solutions_Determinc two lincarly independent solutions to the corresponding homogencous equation Find suitable choice of P, q and f(t) that enables these solutions
All that is known concerning mysterious second-ordcr constant-cocfficient differential equation y" + py f(t) is that t2 + 1 + cos t, t2 + 1 + e' sint; and t2 + 1 + cos t | sin t arc solutions_ Determinc two lincarly independent solutions to the corresponding homogencous equation Find suita...
5 answers
(Smp'ty 3 "Geprantem 1 und, 1 Round [0 enid equaln donrna lexds plucu: 1 rool-liriding prob ern
(Smp'ty 3 "Geprantem 1 und, 1 Round [0 enid equaln donrna lexds plucu: 1 rool-liriding prob ern...
5 answers
Given n 6)-4/+9x 3 , answer thePart 1 of 5(a) Find n ( x)n (5x)9x3Part: 1 / 5Part 2 of 5(b) Find n ().n (1)4 |xl
Given n 6)-4/+9x 3 , answer the Part 1 of 5 (a) Find n ( x) n (5x) 9x 3 Part: 1 / 5 Part 2 of 5 (b) Find n (). n (1) 4 |xl...

-- 0.017963--