Rigid Bodies: Rotation About a Fixed Axis Dynamics (learn to solve any question)

Learn how to solve problems involving rigid bodies spinning around a fixed axis with animated examples. We talk about angular position, angular velocity, angular acceleration, gear ratios, revolutions to rad and much more!
Intro (00:00)
Angular Position (00:24)
Angular Velocity (00:59)
Angular Acceleration (01:25)
Magnitude of Velocity (02:00)
Magnitude of Acceleration (02:57)
Gear Ratios (03:40)
Revolutions to Rad (04:05)
The angular acceleration of the disk is defined by (04:32)
A motor gives gear A an angular acceleration of (06:26)
The pinion gear A on the motor shaft is given a constant angular acceleration (07:55)
If the shaft and plate rotates with a constant angular velocity of (09:05)
Solving cross products:
• How to Solve a Scalar ...
Good website I found for doing cross products:
onlinemschool.com/math/assist...
Find more at www.questionsolutions.com
Book used: R. C. Hibbeler and K. B. Yap, Mechanics for engineers - dynamics. Singapore: Pearson Education, 2014.

[@QuestionSolutions]

At 4:43, in the first example, there is a typo on the diagram. It should be ω = 12 rad/s (angular velocity), NOT v = 12 rad/s (linear velocity). At 6:17, there is a typo, tangential acceleration is 12 m/s^2. Please kindly keep this in mind!

[@Diggydogdraddy]

Man I have been watching the entire dynamics playlist, and reading the comments and all. 1. You reply to almost every single comment. 2. You take your time to make a proper video with proper animations and proper explanaitions. You sir have earned my complete respect, and If i was in the same city as you id physically have to owe you lunch or dinner atleast once. Such a great soul!

[@QuestionSolutions]

Thank you so much for taking the time to write a comment like this, it's really nice of you. I hope the dynamics playlist was helpful and I wish you the best in your endeavors. Really appreciate your comment. ❤

[@Diggydogdraddy]

@@QuestionSolutions I really appreciate your existence. No matter how small the channel is right now, just remember that there are ppl (your watchers and subscribers) that benefit tremendously from your videos. Yes, I have read your comment in one of the videos earlier , you were saying that time is a major issue when the person asked you to tackle other subjects and courses, but no matter how little time you've got, the smallest and the shortest video you make, the closer you get to reachin your goals. reaching more and more people. Just dont give up, i understand how hard it might be to stay motivated relative to having a bigger channel, but you , in my opinion, are doing a great freaking job man!

[@QuestionSolutions]

@@Diggydogdraddy These comments really do make my day. I will keep it going, and I hope you, and many others benefit from these videos, and pass all the courses with flying colors. Thanks again for taking the time to write a comment, I truly appreciate it, and value your insight. :)

[@Diggydogdraddy]

@@QuestionSolutions Thank YOU.

[@QuestionSolutions]

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[@desmondtorevasei]

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[@QuestionSolutions]

You're very welcome! 😊

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[@QuestionSolutions]

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[@TwinklingStar0420]

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[@QuestionSolutions]

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[@austin8313]

With finals coming up the dynamics videos you made are saving my life. I love how much you cover in such a short time and I love the examples you use. My professor uses way too easy of examples when explaining things and so I end up practicing those easy examples and screwing myself over for the exam. THANK YOU SO MUCH!!!!!!

[@QuestionSolutions]

Thank you for your kind comment. I am very glad to hear that these videos helped. I try my best to make them as concise as possible, I know students don't have a lot of time 😅 Anyways, I wish you the very best on your exams and your future endeavors. If you can, please kindly share these videos with your friends/ classmates, it might help them and it'll help the channel a lot too :)

[@alexboudakian2148]

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[@BinodNeupane]

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[@QuestionSolutions]

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[@QuestionSolutions]

Thank you for taking the time to write your comment. Really glad to hear that these videos helped you out. Keep up the great work and best wishes with your studies!

[@axelclear9219]

This is a life saver, thank you so much! I'm taking dynamics online and idk how I would learn without these videos.

[@QuestionSolutions]

You're very welcome! I am really happy that these videos help and wish you the best in your course :)

[@hassanullah3118]

which university ?

[@tacct1kk715]

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[@QuestionSolutions]

I am glad these videos help you out. Keep up the good work and I hope you do great in your course.

[@hugox1106]

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[@QuestionSolutions]

You're welcome :)

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[@QuestionSolutions]

You're very welcome. I am happy these videos are helping.

[@NeoAphelion]

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[@QuestionSolutions]

Happy to help and you are very welcome.

[@Shah.Team-786]

THANKS FOR GOOD EXPLANATION SIR, YOU ARE MY FIRST WHO TEACH ME IN ENGLISH AND I CLEARLY UNDERSTAND IT. ❤❤

[@QuestionSolutions]

I am really glad you liked the explanation. Thanks for watching and I wish you the best with your studies! ❤❤

[@Shah.Team-786]

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[@Caleepo]

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[@QuestionSolutions]

I hope this helps :)

[@mehedihasanrinku2573]

This video deserves a love react

[@QuestionSolutions]

❤ Many thanks!

[@nasseral-kharraz5197]

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[@QuestionSolutions]

I am glad to hear it's making it easy for you, keep up the good work and best wishes with your studies!

[@nasseral-kharraz5197]

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[@QuestionSolutions]

@@nasseral-kharraz5197 You're very welcome!

[@alperyasin710]

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[@QuestionSolutions]

Thank you very much! Really appreciate it.

[@alperyasin710]

@@QuestionSolutions i have a question, at 3:38 a = at +an, but our teacher gave that formula like a=sqrt(at^2+an^2), what is the difference

[@QuestionSolutions]

@@alperyasin710 So what your teacher gave is how to find the magnitude when using the scalar formulas. Each of the "a_t" and "a_n" components can be squared, added together, and then if you take the square root, you end up with the magnitude of acceleration. So that's only for the scalar version, not vector version. For the vector version, you simply add the "a_t" and "a_n" components, but they must be expressed in cartesian form. Not scalar.

[@advait1237]

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[@QuestionSolutions]

I hope you do well on your exam. Best wishes and I am really happy to hear the content is helpful to you :)

[@thomascruickshank7407]

These videos are incredibly helpful. I was wondering if you had an equations sheet for all of the chapters covered?

[@QuestionSolutions]

Glad to hear they are helpful, unfortunately, I don't have an equation sheet, though all of the ones I cover should be in the textbooks you use in your class. 👍

[@thomascruickshank7407]

@@QuestionSolutions No problem! Thank you for all of the help!

[@QuestionSolutions]

@@thomascruickshank7407 You're very welcome!

[@alsix7334]

I think there is a typo at 4:43. You have V0 = 12 rad/s but it should be omega 0 because it is angular velocity instead of linear velocity. You make the best videos. You are literally the only person I watch when I need to understand something from dynamics. No one else has a well put together and easy to follow tutorial series like you. Please keep up the good work. Your videos have helped me so much with my dynamics class. You also have the right voice and speed for these videos! :)

[@QuestionSolutions]

You are absolutely right, it should definitely be ω = 12 rad/s. It is a typo on my part and thank you for pointing it out. I appreciate it. I am really glad these videos help you. And a compliment about my voice, that's a new one 😅Many thanks, and I wish you the best of luck with your studies! :)

[@camerongillespie870]

Thank you, sir.

[@QuestionSolutions]

You're very welcome!

[@amranali60]

Thanks!

[@QuestionSolutions]

Thank you so much for supporting the channel. I really appreciate it! :)

[@maismohammadghannam3910]

Thank you 😊🙏🏻

[@QuestionSolutions]

You're very welcome! :)

[@f.1082]

how can you reply to all comments lmao. you're insane thank you for the amazing videos I will crush my final tonight with your help.

[@QuestionSolutions]

Well, I want everyone to crush their finals, so if they have a question, I try my best to answer. I wish you the absolute best on your exam, keep up the good work! 👍

[@tuhinsuryachakraborty]

very much clarity

[@QuestionSolutions]

I am glad to hear that :)

[@saiprasadsatya3677]

After opening video first I hit like after I watch video bcoz your video never disappoint me

[@QuestionSolutions]

That's really kind of you and I appreciate that. I believe interactions like that cause KZbin to recommend this video more to others, so I am very grateful. Many thanks!

[@programmingprograms726]

Hey, thank you so much for your videos. I'm very curious, do you use some sort of program to change your voice or it's just how you speak?

[@QuestionSolutions]

You're welcome and no, it's just how I speak 🤷

[@yoonmemezaw1206]

You are video is really helpful to me. I would like to know one things that how can we change particle problems to rigid bodies?😇

[@QuestionSolutions]

I am not sure I understand. Particles problems can't be changed in to rigid bodies because particles are particles, and rigid bodies are rigid bodies. Usually when you get angular velocity, or angular acceleration, you're dealing with rigid bodies. :)

[@sadratehrani415]

Hi, just a question about the first example. My course hasn't gone into integrals as of yet so I tried to solve for the final angular velocity using the angular rotation equations. If angular acceleration is equal to 3t^2 + 12 then inputting t=2 into that give the angular acceleration (24rad/s^2). then I used wf=wi + angularacceleration*t. however that gives a wf value of 60. Where am I going wrong here? Thank you in advance

[@QuestionSolutions]

Please provide a timestamp so I know where to look. If integrals haven't been covered, please use symbolab or wolfram alpha to see the steps.

[@chrisrider5770]

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[@QuestionSolutions]

Thank you!

[@lolhavefun9201]

You saved me during midterms. Here I am, back again for finals lol

[@QuestionSolutions]

I wish you the best on your finals! 👍👍👍👍

[@lolhavefun9201]

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[@QuestionSolutions]

@@lolhavefun9201 Awesome!!! Well done :) Now I wish you the best on your other courses and your future endeavors.

[@vh4yzs3r]

thanks

[@QuestionSolutions]

You're very welcome!

[@AntoninoSalvia]

Not sure it was already mentioned but on question #1 for the Tangential acceleration I think there is a typo. Shouldn't it be 12 instead of 10? Also I really appreciate your videos they have helped me so much!

[@QuestionSolutions]

Could you please provide me with a timestamp to the location you're referring to?

[@zuhair95]

Many Thanks ... At 01:58 u=you said the rectilinear equations when it is constant acceleration . but why you mentioned the projectile equations only !?

[@QuestionSolutions]

Projectile motion uses rectilinear equations, so I am unsure of your question? On the blue box is the new equations that use constant angular velocity, on the right, in the red box are the rectilinear equations used in projectile motion and a host of other problems with constant linear velocity. 👍

[@zuhair95]

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[@QuestionSolutions]

@@zuhair95 Many thanks! ❤

[@johnhartney7576]

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[@QuestionSolutions]

Thank you very much!

[@godfredo2275]

in hindsight of the last question, could you have used the position vector of OB?

[@QuestionSolutions]

We don't know where point B is, so we can't use it.

[@pecsamridhisharma8789]

At 6:17, in Ques 1, angular tangential acceleration should be 12m/s^2.

[@QuestionSolutions]

Yes, there is a typo 😅

[@r2k314]

can you explain how the results of the last example make physical sense? There are components of velocity, and normal acceleration in the k direction.

[@QuestionSolutions]

It just means the velocity has 3 components. So one way to think about it, is to understand that we live in a 3 dimensional world. That means we can have components for any three axes. For example, when we apply a force to a wrench, that creates a moment. Assume it's a counter-clockwise moment. If we use the right hand rule, we would get a vector that points straight upwards, even though we are just turning something counter-clockwise. Most of these things are useful when doing math, or for further explanations.

[@r2k314]

@@QuestionSolutions Ok. Thank you! But I thought the tangential and centripetal accelerations would be confined to the plane of the plate.

[@QuestionSolutions]

@@r2k314 Please see: hyperphysics.phy-astr.gsu.edu/hbase/rotv.html

[@mohdyash636]

In last question 9:09 Why you multiplied w with posiition vector of OA to get w??

[@QuestionSolutions]

It's the procedure to write a force in cartesian form. It's usually covered in statics (the course before dynamics).

[@studyhard6214]

Hello, I have a question. At 11:00, I thought we could use the formula "-ω²r" to find the normal acceleration, but this way I find a different result. How can it be?

[@QuestionSolutions]

All the values are in cartesian form, the equation you're using is a scalar one. We need our answer in cartesian vector form, so you have to use a cross product to get the answer.

[@studyhard6214]

@@QuestionSolutions Actually, I needed to put an arrow on r to represent the vector, but I couldn't on the keyboard. So this formula is actually in Cartesian form, isn't it? And the question 16.37 in the book is solved as I said. In that question, w = 5j, so it consists of one component, could it have something to do with it? I was glad to see that this was a bit shortcut, but I am confused now. Also, thank you for really quick response!

[@QuestionSolutions]

@@studyhard6214 When you have 1 component, it is pretty much just a scalar value. Think about it this way, if we find the magnitude of our position vector, we get the scalar value. When it's a single component, so let's say 5j, the magnitude is just 5. This is why you're getting the same answer, however, adding other components will not yield the same results. I also think it's easier if you do any question the full way because it allows you to see the small differences like these, and it becomes second nature. Please see: 3:23, what you're doing is trying to find vector values using scalar equations. 👍

[@studyhard6214]

@@QuestionSolutions Got it, thank you for your explanatory and quick answers! 😇

[@QuestionSolutions]

@@studyhard6214 You're very welcome. Best wishes with your studies :)

[@kazimozel6193]

yur a saint bro!!

[@QuestionSolutions]

👍 Good luck with your studies!

[@MuhammadQasim-kw1er]

Which softwre you are using?

[@QuestionSolutions]

I use after effects to animate.

[@abdallahindimi3484]

at 7:04 when we are isolating for the angular velocity, cant we just divide 15^2/2 which is 112.5 or multiply both sides by 2 twice? just a bit confused

[@QuestionSolutions]

You can do it however you like, whatever is easier for you :)

[@abdallahindimi3484]

it doesn't give me the same answer when i use 112,5😬, so im probably doing something wrong @@QuestionSolutions

[@QuestionSolutions]

@@abdallahindimi3484 Could be a numerical operation error? 😅

[@abbybeew2996]

@questionsolutions how did you get .004theta at 7:06

[@QuestionSolutions]

You multiply 0.002 by 2 to get rid of the fractions on left side.

[@kevinshibu7939]

Hi, Regarding the first question, I have a query. I tried the question using my own method and my module formula sheet. However I did not get the answer you seemed to get. These were my steps: Formula: Omega = Omega0 + alpha x (time) - for constant acceleration Omega0 = 12 Alpha - sub in t=2 to get 24, multiply by 2 to get (alpha x time) = 48 48 +12 = 60 for omega use (v= omega x r) and get v = 30 I can't seem to understand why this does not work. I want to guess it's because of my use of the formula? Would you be able to explain why this is incorrect?

[@QuestionSolutions]

Interesting question. You assumed a constant acceleration when that isn't true for this question. When you plugged in your time, you assumed a linear graph, which again isn't true for this problem. So there really is no way of getting around the integration part for this problem. Your method would work if the acceleration is constant, but we're given a quadratic equation for the angular acceleration.

[@kevinshibu7939]

@@QuestionSolutions I see. Thank you for clearing this up for me, i understand why it didn't work now!

[@QuestionSolutions]

@@kevinshibu7939 Awesome! Keep up the good work and best wishes with your studies :)

[@wakeawake2950]

Nice video , but can you tell me why the direction of angular velocity is perpendicular to the plane of motion? Plzz reply🙏🙏❤️❤️

[@QuestionSolutions]

When you have a rigid body, let's say it's a simple bar that's attached to a fixed point, and it's rotating about that point, the velocity vector will always be perpendicular to that bar. The reason is that if you look at the edge of the bar rotating, you will see that it's traveling along a curve. That means the velocity vector will be tangent to that path. Please see this video from the time (0:43): kzbin.info/www/bejne/Z5K0aniQfLKXes0 There I show the path and how velocity is tangent to that path. So when a rigid body rotates about a fixed point, the velocity is tangent to the curve, or perpendicular to the body.

[@wakeawake2950]

@@QuestionSolutions thank you so much, this cleared my doubt and keep making these videos, it helps alot🤗🤗👍

[@QuestionSolutions]

@@wakeawake2950 Really glad to hear that! Best of luck with your studies 👍

[@islahizham]

@@QuestionSolutions wow you explain it in the easiest way! i clearly understand it from the first reading

[@QuestionSolutions]

@@islahizham I am really glad to hear that :)

[@ElCrankoPunko]

@6:32 the gear gear A has initial angular velocity & how come it has initial angular position as zero or do we take that initial position as origin that’s why zero

[@QuestionSolutions]

That is correct, we take that as the initial position.

[@ElCrankoPunko]

@@QuestionSolutions thank you

[@deyardeyar8163]

so in 5:28 we find the final angular velocity or the magnitude of angular velocity ?

[@QuestionSolutions]

Angular velocity is represented with "ω", and linear velocity is "v". It's just the magnitude of velocity, so it's represented with v. No magnitude of angular velocity, just magnitude of velocity.

[@sjfn9846]

At 7:03 how does the upperbound of the integral become (w^2)/2 because if I integrate w it will be [(w^2)] and then inserting the bounds from w to 15 will give me ((w^3)/2) - ((15^2)/2). I would appreciate if you could answer my question.

[@QuestionSolutions]

You are doing integrals incorrectly. The integral of x, is x^2/2. So here, the integral of w is w^2/2. See www.symbolab.com/solver/integral-calculator/%5Cint%20xdx?or=input I encourage you to review integrals because they are a key part of dynamics and will be used a lot :)

[@sjfn9846]

@@QuestionSolutions Hi sorry I forgot to write the division by two, of course the integral of w is (w^2/2) but my question was after the integration is done the input of of 0 to w which are the intervals of the integration. Because [w^2/2] with the insertion of w would be (w^3/2).

[@QuestionSolutions]

@@sjfn9846 So you are solving definite integrals incorrectly. So let's say we solve an integral, and we have x^2/2 and the lower and upper bounds are 3 and 2. In that case, we plug in (3^2/2-2^2/2). Now, instead of 3 and 2, we have x and 2. Again, all we do is just plug them in. So we get (x^2/2-2^2/2). It doesn't become x^3/2-2^2/2. Here is a simple example: www.symbolab.com/solver/definite-integral-calculator/%5Cint_%7B2%7D%5E%7Bx%7D%20xdx?or=input I hope that helps :)

[@sjfn9846]

@@QuestionSolutions I got it now thank you so much for such a nice channel and the fast replies, you’re amazing at describing this to such an easy level.

[@QuestionSolutions]

@@sjfn9846 You're very welcome. I just want to make sure, you understand how to do the limits on a definite integral now? If not, please let me know. Otherwise, best wishes with your studies!

[@orihimeinoue41]

Hello can you answer this question I'm not sure about my answer. Does a rigid object in uniform rotation about a fixed axis satisfy the first and second conditions for equilibrium? Why? Does it then follow that every particle in this object is in equilibrium? Explain

[@QuestionSolutions]

Please ask these types of questions not relating to the questions shown in videos with your professor/ TAs during their office hours. They can offer a much better explanation in person.

[@manishjoshi2821]

Which application do you use please tell me 🥺

[@QuestionSolutions]

Do animate, it's After Effects, to draw diagrams, its Illustrator.

[@jarnobreur]

At 7:04 why is everthing on the left side/2? and how did the right side go from 0.06 to 0.002? My head cant figure out why. Thanks!

[@QuestionSolutions]

So this is just the process of solving an integral. When you raise the variable to a power, you divide by that value. On the other side, θ was squared, so you raise it a power, so you get θ cubed, and then divide it by 3. 0.006/3 = 0.002. If integrals weren't covered in your courses, then you probably won't get any questions with them. If they were covered in your courses, please take the time to review them, it'll make your life a lot easier :)

[@jarnobreur]

@@QuestionSolutions Alright, thanks alot for your response!

[@QuestionSolutions]

@@jarnobreur You're very welcome!

[@Firas_Kay]

the best

[@QuestionSolutions]

Thank you!

[@floof505]

at 11:05 Shouldn't Vc's j component be +3.6j instead of -3.6j? That's what I got

[@QuestionSolutions]

No, it's negative. You made some error in positives and negatives. Also, here is a good website to double check your work: onlinemschool.com/math/assistance/vector/multiply1/

[@floof505]

@@QuestionSolutions I just realised where I went wrong. Thank you!

[@QuestionSolutions]

@@floof505 You're very welcome!

[@WahibAlaoui]

6.48There is any possibility to calculate like this ?? Alpha=dw/dt W(t)= 2t +0,006Q² to +15 W= 40 + 0,006 (20)³ +15..Thank you in advance

[@QuestionSolutions]

I don't think you get the same answer with your method. Also, are you trying to integrate with respect to "t" with an equation that has theta? I am a bit confused with your notation, is "Q" suppose to be theta and "t" time?

[@WahibAlaoui]

@@QuestionSolutions ​i have just one question why don't you use this formula α =dω / dt ? thank you

[@QuestionSolutions]

@@WahibAlaoui The equation we are given for angular acceleration is with respect to theta, the change in angle, not angular velocity.

[@ahmedfekry4810]

Not sure but isnt the angular displacement for c at 9:02 should be = Wt which is 1.68 * 2 ??

[@QuestionSolutions]

So we are using gear ratios to find the rest of the unknowns after we found ωA and θA. Notice how we found those values at t = 2 seconds (8:15). That means the 2 seconds were already accounted for. I hope that helps :)

[@ahmedfekry4810]

@@QuestionSolutions thanks for your help

[@QuestionSolutions]

@@ahmedfekry4810 You're very welcome!

[@Simrannnn0006]

Hiilo hii , actually m a jee aspirtant of 2023 , m having a lot backlogs of class 11, everyone was saying that for making my 12 better i need cover some imp topics of class11 ,they told me to do fixed axis rotation the basic concept i just want know is it the concept of fixed axis rotation u explain????

[@QuestionSolutions]

Yes, this video is about fixed axis.

[@Simrannnn0006]

@@QuestionSolutions ok Thank u

[@QuestionSolutions]

@@Simrannnn0006 The whole playlist should cover most of the stuff you need. 🌟

[@Simrannnn0006]

@@QuestionSolutions thank u 💓

[@DraftingCADemy2023]

The first problem isn't a constant angular acceleration type of problem right? Which is why I can't arrive at the correct answers using the constant acceleration equations.

[@QuestionSolutions]

No, angular acceleration is given as an equation that depends on time.

[@I3MVR]

At 5:24, the W should be 32, not 44, becasse when you do the derivative, you will get 6t, and when you apply the limits, you will get 24, not 32

[@QuestionSolutions]

I am not sure I understand what you're saying? w-12=32 ==> w=12+32 ==> w=44 The integral is also correct. You're making a numerical error but I don't know where since I can't see your steps. www.symbolab.com/solver/definite-integral-calculator/%5Cint_%7B0%7D%5E%7B2%7D%203x%5E%7B2%7D%2B12%20dx?or=input

[@yakupcaferdemir3116]

Thank you very much

[@QuestionSolutions]

You're very welcome!

[@thehungarywaffleinc.7775]

i love you

[@QuestionSolutions]

❤

[@MrPJBarney]

Thank you so much! God bless you. Jesus loves you.

[@QuestionSolutions]

You're very welcome ❤

[@hayaalawwad]

shoouldnt the angular acceleration .5x24=12?

[@QuestionSolutions]

Yes, if you are referring to 6:24, the tangential acceleration is 12 m/s^2, I made a typo there. Thanks for pointing it out.

[@darrylcarter3691]

Wait I thought anything time 0 is zero. where did the -3.6j come from?

[@QuestionSolutions]

Please use time stamps so I know which part you are referring to. To answer your question, it's the multiplication of 12 by 0.3 that gives you 3.6. Please see: kzbin.info/www/bejne/fGmseaWdaNWWbck

[@eksp.1]

7:01, how it became 0.002...? btw thanks :)

[@QuestionSolutions]

0.006/3 = 0.002 don't forget to solve the integrals properly :)

[@ziyadgames5340]

6:59 why angular position =0 ? ( theta)

[@QuestionSolutions]

Think of it as the initial "location" so it's set to zero and counted from that point on.

[@10001willy]

At 6:15, we see that 24(0.5)=10, but the actual answer should be 12

[@QuestionSolutions]

Yes, it's already mentioned on the pinned comment :)

[@10001willy]

@@QuestionSolutions oh I didn't see that part. Good to know, thank you!

[@QuestionSolutions]

@@10001willy It's okay, best wishes with your studies!

[@prasannapratapthapa2744]

My mechanics exam is tomorrow

[@QuestionSolutions]

Best wishes on your exam!

[@seranvishwa8714]

Watching 3 hrs brfore the exam 😅😅

[@QuestionSolutions]

Best wishes with your exams :)

[@seranvishwa8714]

@@QuestionSolutions thank you and thank you for your video. It is really good and clear👊❤️💫

[@QuestionSolutions]

@@seranvishwa8714 That's awesome to hear! Let me know how your exam went 👍

[@seranvishwa8714]

@@QuestionSolutions Okay Definitely Bro🤜👍

[@seranvishwa8714]

@@QuestionSolutions Exam was good Thank you🤜

[@chenmoney1920]

haha , funny at 6:23 , 24*0.5 = 10 ????? haha is 12

[@QuestionSolutions]

Thanks for catching the typo 😅

[@eyadelsayed4799]

I didn't like some of your solution .

[@QuestionSolutions]

Okay, thanks for the feedback 👍