He takes a tripe integral because the cylinder is three-dimensional, and we need to take into account all the mass. That's why we're using one integral for each dimension (z, r, and theta in this case).
chain rule is when you have for example in the case of a cylinder dm/dv = density and you know that v=π r^2 h so to get dm in terms of dr and density u differentiate v so you get dv/dr = 2πrh which gives dv= 2πrh dr then you substitue dv in 1st equation to get dm= density * (2πrh dr) then you integrate I= ʃ r^2 * [density *(2πrh dr) ] using limits from 0-R (radius of uniform cylinder) and then factorize to get 1/2 *MR^2
no. density = mass /volume. density = 1. Therefore, 1 = mass/volume and mass = volume. The volume of the sphere is pi*h*b^2. Therefore mass = pi*h*b^2.
You may take arbitrary density, if you will. In this case he makes density to be equal to 1 because it is convenient (we do not need to write additional coefficient (delta in the case) before dV. We have dV = dm but actually (in general) it is delta * dV = dm).
This is the centroidal moment of inertia. Things get bizarre when you find the inertia of an object that is at a distance from the axis. Still! Awesome job Professor Lewis. :-)
Great video, I have a question though. Can anyone explain why the inertia of a disc difffers from the inertia of the cylinder? Is my assumption that a disc being a shortened cylinder wrong? Thanks.
Dunno if it's too late but it does not differ from the moment of inertia of a disk, the moment of inertia of a disk is 1/2 M r^2 and the derived form here is 1/2 M b^2, where b is the radius of the cylinder.
hello Joel, this is a good video, can i use the same method to find the moment of inertia of a hollow cylinder. i.e, find the moment of inertia of inner minus outer volume using the above method of triple integral.
I am a mechanical engineer. I have studied a course called Mechanics of materials. Torsion part is dealt with 'area moment of inertia', which is a slight variation of mass moment of inertia. I remember polar moment of inertia(Izz) is one-half of the regular moment of inertia( Ixx or Iyy).. So, I am pretty much sure I am right
+Leopoldo Madiam He chose it that way so it could be computed more easily. In real-life situations, Objects don't usually have same density everywhere since they're probably made up of more than one element. But well, the main point of the video is the idea behind the computation and not doing something that complicated. Sorry for bad english n_n
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He takes a tripe integral because the cylinder is three-dimensional, and we need to take into account all the mass. That's why we're using one integral for each dimension (z, r, and theta in this case).
chain rule is when you have for example in the case of a cylinder dm/dv = density and you know that v=π r^2 h so to get dm in terms of dr and density u differentiate v so you get dv/dr = 2πrh
which gives dv= 2πrh dr then you substitue dv in 1st equation to get dm= density * (2πrh dr)
then you integrate I= ʃ r^2 * [density *(2πrh dr) ] using limits from 0-R (radius of uniform cylinder) and then factorize to get 1/2 *MR^2
no. density = mass /volume. density = 1. Therefore, 1 = mass/volume and mass = volume. The volume of the sphere is pi*h*b^2. Therefore mass = pi*h*b^2.
I love your delivery dude! You examples are great!!! Thank you!:
You're right, I overlooked at the fact that we are given density=1.
Very, very clear.
You may take arbitrary density, if you will.
In this case he makes density to be equal to 1 because it is convenient (we do not need to write additional coefficient (delta in the case) before dV. We have dV = dm but actually (in general) it is delta * dV = dm).
Excellent video
Great explanation, Joel!
This is the centroidal moment of inertia. Things get bizarre when you find the inertia of an object that is at a distance from the axis.
Still! Awesome job Professor Lewis. :-)
The walk away as we pause the video.
I didn't pause the video or try to find a solution. Full disclosure.
Go to Jail! Do not pass GO, do not collect 200!
and why is Dv=r.dz.d (theta) ?
Great video "shocker right" :)
Very good video. Thanks!
I like this guy.
Gracias
great!
Great video, I have a question though. Can anyone explain why the inertia of a disc difffers from the inertia of the cylinder? Is my assumption that a disc being a shortened cylinder wrong? Thanks.
Dunno if it's too late but it does not differ from the moment of inertia of a disk, the moment of inertia of a disk is 1/2 M r^2 and the derived form here is 1/2 M b^2, where b is the radius of the cylinder.
what if the cylinder is non-uniform, i.e. [delta] = [constant] * r / b ?
shouldn't the answer be with M/density not only M
what about hemisphere. can we calculate. using cylindrical coordinate
1st ._. but we didn't take the triple integration part we learned chain rule
THANK YOU
Ok...now find moment of inertia in respect to x or y axes (bottom of cylinder) without using any integrals.
Someone plz tell me is inertia matrix is invertible?
but, why does he take the triple integral from r^2? i mean what's the logic?
hello Joel, this is a good video, can i use the same method to find the moment of inertia of a hollow cylinder. i.e, find the moment of inertia of inner minus outer volume using the above method of triple integral.
why is the density equals to 1?
Prep school for Mechanical or Industrial Engineering. what semester do you take it?
how to calculate moment of inertia about cylinder's horizontal axis
perpendicular axis theorem... Izz=Ixx+Iyy..
Ixx=Iyy. so, Ixx=Iyy= 0.5(Izz). hope it helps :)
I am a mechanical engineer. I have studied a course called Mechanics of materials. Torsion part is dealt with 'area moment of inertia', which is a slight variation of mass moment of inertia.
I remember polar moment of inertia(Izz) is one-half of the regular moment of inertia( Ixx or Iyy)..
So, I am pretty much sure I am right
3rd semester? Why what are you majoring in?
what year do you take this course ? I took it 3rd semester in france
11th grade in india.
wtf is chain rule ?
you can easily do this with a single integral- seems redundant to do it with three just because you can.
This guy is the worst out of all the GSI on this series. Worst. He’s now at GWU as some mediocre instructor.
Why is the density equals to one?
+Leopoldo Madiam He chose it that way so it could be computed more easily. In real-life situations, Objects don't usually have same density everywhere since they're probably made up of more than one element. But well, the main point of the video is the idea behind the computation and not doing something that complicated. Sorry for bad english n_n