How to Find Mass Moment of Inertia | Mechanics Statics | (Solved Examples)

Learn to find the mass moment of random objects, composite bodies, and learn to use the parallel axis theorem. We go through multiple examples using the shell and disk method step by step, and cover everything from finding the volume to writing the mass moment of inertia in terms of mass.
Integral fully solved step by step: bit.ly/3u4TwC1
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Intro (00:00)
Parallel Axis Theorem (02:29)
Determine the mass moment of inertia of the cylinder (03:35)
The right circular cone is formed by revolving the shaded area (06:18)
Determine the moment of inertia Ix of the sphere (08:59)
The slender rods have a mass of 4 kg/m (10:15)
The thin plate has a mass per unit area of (11:35)
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Book used: R. C. Hibbeler and K. B. Yap, Engineering Mechanics Statics.
Hoboken: Pearson, 2017.

[@joeyGalileoHotto]

I just finished Dynamics and took the final exam last week, and you helped me a lot man. Thanks.

[@QuestionSolutions]

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[@QuestionSolutions]

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[@joeyGalileoHotto]

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[@pure_ligma2387]

@@joeyGalileoHotto damn. how did you study for it? Any tips or tricks? I'm currently reading hibbeler and only going through examples and preliminary and fundamental exercises and watching these videos. I have about 8 days left to study before the exam.

[@kerenhappuch5605]

I love how he presents a whole subtopic in a few minutes 😊

[@QuestionSolutions]

I hope it was helpful to you 🙂 I wish you the best with your studies, and keep up the good work!

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[@QuestionSolutions]

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[@QuestionSolutions]

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[@QuestionSolutions]

Thank you very much for your very kind words. While I am sure these videos helped you out, most of it was all you and your hard work. Well done and I wish you the best with your future studies. :)

[@natte431]

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[@theazndonut1]

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[@QuestionSolutions]

😅 Glad to hear they've been helpful. I wish you the best on your FE exam!

[@georgemikhaiel2620]

Your videos are amazing! My uni has a couple chapters after shear force on Dry Friction, Wedges and sipping/tipping...It would be awesome if you could make videos on those topics too!! Im coming back next semester to watch you Dynamics Playlist!!

[@QuestionSolutions]

Usually, first year courses don't cover those topics, which is why I skipped them for now. Over time, I will come back and fill those in for the statics playlist :)

[@georgemikhaiel2620]

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[@QuestionSolutions]

Glad to hear it :)

[@mahajaved207]

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[@QuestionSolutions]

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[@natte431]

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[@QuestionSolutions]

Thank you very much. Maybe in the future, there will be another playlist that can be helpful to you :)

[@sairajgurav6680]

Nicely explained!! Thanks

[@QuestionSolutions]

You're very welcome! Best wishes with your studies.

[@xanonymousx6014]

I really appreciate your great work bunch thanks, and also if you can do fluid mechanics that will be awesome

[@QuestionSolutions]

I will add that to my list :)

[@JesusMartinez-zu3xl]

this helped a lot! thank you

[@QuestionSolutions]

I'm glad this helped you. Best wishes with your studies!

[@kimbrodersen5075]

A little something for the algorithm, great video!

[@QuestionSolutions]

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[@wt7146]

Your channel is a life saver! I took my statics exam last week and I owe whatever grade I get to you! Do you think you would be able to do differential equations as well?

[@QuestionSolutions]

Glad to hear! I hope you did really well :) I will add differential equations to my list, though it probably won't get to the top for a bit. I'll do my best! 😅

[@gwv8647]

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[@QuestionSolutions]

@@gwv8647 Thank you :)

[@carlosnjiri3679]

am greatfull you really good at explaining.

[@QuestionSolutions]

Thank you very much, I really appreciate it. Keep up the good work!

[@broz2308]

Truly excellent videos man, big appreciation. My course of study uses the same book from hibbler too. Will you continue on other topics such as friction and center of gravity?

[@QuestionSolutions]

You're very welcome. I might continue the statics lessons in the future. I tried to cover as much first year material as possible. 👍

[@broz2308]

@@QuestionSolutions looking forward, take care.

[@samkeloinnocent8316]

Thank you so much😊

[@QuestionSolutions]

You're very welcome! :)

[@darrylcarter3691]

I remember this useful in structural engineering

[@QuestionSolutions]

And very useful for dynamics 👍

[@HashemAljifri515]

I took this topic today in engineering dynamics, I took it in statics and physics 101 and currently taking it in mechanics for pure bending. and your explanation is wonderful, but this topic is tedious; cause there are different formulas for the same object in different axises :(. And finding inertia by integrating is a bit harder lol. I hate to say it but this topic is tough 😂

[@QuestionSolutions]

It's definitely on the more difficult side of things for students, especially if they need to do integral. But as with most topics, the more problems you solve, the easier it gets. You got this! Keep up the great work and do your best :)

[@mountainyatri]

This is very nice and informative video.....and ur voice 😁😁😁

[@QuestionSolutions]

Thank you, I am glad it was informative :)

[@mountainyatri]

@@QuestionSolutions actually I'm facing a lot problem in finding moment of inertia and solving all those questions...I I'm like good please help me bcz my entrance exam is coming.....pls help me through this....else u upload video of all the questions related to the topic...soon as possible It will be grateful ...

[@QuestionSolutions]

@@mountainyatri There aren't anymore videos to upload about this topic :( I covered everything required to solve for the mass moment of inertia. There are also other videos on youtube about the subject, maybe they can offer better insight for you? Sorry I couldn't be more helpful :(

[@mountainyatri]

@@QuestionSolutions no problem:-) ..this is much u can do.. I'll finalise my doubt and queries.. Thanks for ur genuine reply.

[@QuestionSolutions]

@@mountainyatri You're very welcome. Best wishes on your upcoming exam!

[@kaarthik9]

thanks mate

[@QuestionSolutions]

You're welcome!

[@vaibhav8009]

Great BRO ....PLEASE PREPARE FOR THERMODYNAMICS ALSO

[@QuestionSolutions]

👍

[@samkeloinnocent8316]

@@QuestionSolutions Do you have Thermos yet?

[@QuestionSolutions]

@@samkeloinnocent8316 Yes, there is a whole new playlist, getting updated every week, or as fast as I can make them. kzbin.info/aero/PLXePpKFSUW2bcuUy2fKCI9gKJJ5GXjg92

[@MaximeBellis]

Very helpful videos, what’s your next topic?

[@QuestionSolutions]

Either thermodynamics or circuits. 👍

[@MaximeBellis]

@@QuestionSolutions circuits ❤️

[@QuestionSolutions]

@@MaximeBellis I will add that to my tally :)

[@kursatper5912]

Thanks

[@QuestionSolutions]

You're very welcome!

[@ndndsnsnna433]

Hey , I have one question, how would you calculate the moment of Inercia in the last example, if one of the circles was a bit to the left ( lets say 0.1 m from the z axel)?

[@QuestionSolutions]

Pretty much the same way, since if it was about the y-axis that the inertia was calculated, we just need the distance from the y-axis.

[@darrylcarter3691]

Do one on finding the radius of gyration and Center of Gravity

[@QuestionSolutions]

I probably won't do one on that. It's just using a single formula and for most problems, it's just plugging in values and will be given to you. Also, I think I am going to stop with statics now and move onto a different topic :)

[@Lucy-gs5rm]

Hello sir, how would you solve the last example when you should calculate it for around the z-axis?

[@QuestionSolutions]

Pretty much the same way, but you need to use the mass moment of inertia equations about the z axis instead of the y axis. Also, the distances should be to the z axis. You can find the equations online.

[@zerilioner639]

Hi, I have one math question: Two particles have positions at time t given by S1=4t-t^2 and S2=5t^2-t^3. Find the velocities V1 and V2 at the instant the accelerations of the two particles are equal.

[@QuestionSolutions]

Hi, I don't really solve problems like this because then I have to solve everyone's problems and it will take too long 😅 Your professor should have their own allocated office hours. Please use them, they are super helpful and most professors sit idly in their office waiting :)

[@arindammandal9378]

Differentiate twice wrt t you get accelation as a function of time than equate them you will get required time .now first derivative wrt t you get respective velocities

[@hummustahina5429]

moment of inertia of last part of the thin plate must be 1/12m(a^2+b^2) since we want it in the middle of the plate right? what you did is getting the inertia on a side

[@QuestionSolutions]

We want it about the y-axis, not the middle.

[@avery-4223]

I'm at the end of my statics course and next semester I'll have to take solid mechanics, does anyone have any reccomendations for another youtuber? i cant seem to find any solid mechanics videos on this youtube channel.

[@QuestionSolutions]

I don't have any recommendations but I am sure there are lots of great videos on KZbin :)

[@saiprasadsatya3677]

2:22 , I am confused in writing dv ,dz , sometimes I write pie*dy^2 please explain how we can find variables in small element

[@QuestionSolutions]

It's based on the direction/axis you're integrating. It's up to you which way you want to do it, you just need to make sure your equation is setup for that direction (x, y, z).

[@jackgoris7602]

I dont get when you need Ix or Iy or Iz because sometimes they have different mass moment of inertia and i dont know wich one i need

[@QuestionSolutions]

So that depends on the direction you're going to integrate. For example, if you decide to integrate over the x-axis, then you'd have Ix, if it's the y-axis, then Iy, etc.

[@astrobiology6218]

Great video but I dont understand why we have to use the extra step in the disc method. I = int(r^2 rho dV). When we know rho and dv=pi y^2 dz why can't we substitute that dv directly? If we do that we are missing a factor of 0.5 from the moment of inertia of the disc. But why is what I describe above wrong?

[@QuestionSolutions]

This is more to do with calculus than dynamics, but the essence of it is that with the shell method, the integration is done perpendicular to the axis where as with the disk, it's integrated parallel to the axis. This means for the mass moment of inertia (when we consider density), you have to take an extra step. This "step" is actually just finding the mass moment of inertia for a disk (perpendicular to the axis), and then adding all of them up using an integration (because the volume is hollow). If you have time, I encourage you to look at the 2 methods of integration in your calculus book. This will give you the proof for both methods and their shortcomings. If you don't have time, then forget all of this and just remember the extra step with the disk method vs the shell method. It's hard to answer without actually drawing it out and showing it to you on paper, so if you're still really curious about it, please visit your professor during office hours.

[@user-oz5yk9bm5c]

what does the d in dV, dZ, dY and dr mean?

[@QuestionSolutions]

So this is from calculus. In very simple terms, the "d" is delta, so an infinitesimal change. If we say "dx" then that's an infinitesimal change in x. If we say dV, then that's an infinitesimal change in V.

[@programmingprograms726]

I've gone thru this video 3 times now and I thought I understood each time but each time I try out a problem I get stuck! Please solve at least 2 questions from the text book to give us an example.

[@QuestionSolutions]

Keep trying, don't give up. Try to solve the problems I show on the video (without just memorizing the steps). If you get stuck, see what happens next in the video. I think I've solved 5 problems in this video :)

[@bigbluexxxx2705]

How come you didn’t cover friction and center of gravity and centroid?

[@bigbluexxxx2705]

( in regards to all your videos of mechanic statics?)

[@QuestionSolutions]

My faculty does not cover those subjects in a first year course. So I wanted to finish other subjects to cover more stuff for first year students.

[@yuvarajucherukuri4753]

hllo sir

[@QuestionSolutions]

Hello!

[@matteoderibaucourt7682]

why is the mass moment of inertie from the disc 1/4… and not 1/2…

[@QuestionSolutions]

Please give a timestamp so I know where you're referring to. Thanks!

[@shaunbabar]

pls teach some trick to find it

[@QuestionSolutions]

There are no tricks. If you do a lot of questions, you will gain a firm understanding of it.

[@brainloading5543]

You made a minor mistake that confused me a bit. The mass is equal to the volumic mass times the volume. The density, in itself, has no unit. So multiplying it with a volume would just get you another volume

[@QuestionSolutions]

Could you tell me where you're referring to?

[@brainloading5543]

@@QuestionSolutions 0:57 "we represent density with the Greek letter ro." but what you are actually representing is the volumetric mass, not density

[@QuestionSolutions]

@@brainloading5543 Not sure if you're confusing terms or maybe it's not clear enough on my part in the video. Here, it is indeed density we are representing with the Greek letter rho. Density is sometimes also called volumetric mass density. It does have units, the SI units being kg/m^3. Mass can be found by multiplying density by the volume. Whenever you have doubts, multiply the units out and it will help. So here, we have m = ρv, so the units would be (kg/m^3)(m^3) => kg. So you end up with mass. I hope that makes sense. If you need further reading, please refer to your textbook, or see: www.engineeringtoolbox.com/density-specific-weight-gravity-d_290.html

[@suleymantnr6749]

where is the centroid topic?

[@QuestionSolutions]

If it's not in the playlist, then I didn't cover it 😅

[@oscarsvensson5277]

Can someone please do a ”mass moment of inertia” counter? My guess is that he said it 134 times 💀

[@QuestionSolutions]

Considering that's the topic being discussed, I guess I should have called it something else? 🤣🤣

[@henriettasmith7096]

Hi, I have a very difficult question, I am wondering if you can help guide me into answering this. Can I email you for help?

[@QuestionSolutions]

I don't really solve questions like that since I have to solve everyone's questions and it'd take a really long time to do. 😅 But you can email me and I'll take a look. email is contact @ questionsolutions.com

[@henriettasmith7096]

@@QuestionSolutions Ah ok, that's fine then. I emailed it, but no worries. :D