How Do You Calculate the Moment of Inertia for a System of Particles?

[richardnumber]

Hey everyone,

Im solving a group of questions based on a system of particles, A(2,0) with a mass of 4kg, B(0,1) with a mass of 6kg and C(1,1 ) with a mass of 3kg. I have to find the centre of mass and also calculate the moment of inertia.

To find the centre of mass I am attempting the following:

" 1/13 [(4.2) + (6.0) + (3.1)]= 3/13 "

" 1/13 [(4.0) + (6.1) + (3.1)]= 1 1/65 "

so centre or mass wud b? ... (3/13, 1 1/65 )

Im not sure if this is correct?

Also I am unsure of how to calculate the moment of inertia?

I really would appreciate any help,

Regards,

Richard

 

[Doc Al]

" 1/13 [(4.2) + (6.0) + (3.1)]= 3/13 "

I assume you mean:
1/13 [(4*2) + (6*0) + (3*1)]= ??

That would be correct for the x-component, but your answer is not. Redo your arithmetic.

" 1/13 [(4.0) + (6.1) + (3.1)]= 1 1/65 "

Similar comments for the y-component.

Also I am unsure of how to calculate the moment of inertia?

The moment of inertia of a particle about some axis is mR², where R is the distance to the axis. Where's your axis?

 

[richardnumber]

oops! few mistakes there, yeah the correct answer would be (11/13, 9/13) ?? My axis is the centre of mass. how would I apply this to the system in this equation?

thank you so much for your help btw!

- Richard

 

[Doc Al]

oops! few mistakes there, yeah the correct answer would be (11/13, 9/13) ??

Good.

My axis is the centre of mass. how would I apply this to the system in this equation?

Start by finding each particle's distance (or distance squared) from the center of mass. Then calculate the moment of inertia of each particle. Add them to get the total for the system.

 

[richardnumber]

thanks again for your quick reply! really helpful! :D

- for the length of each particle to the centre of mass I am using:

d = *squareroot* [ (a-x)^2 + (b - y)^2 ]

where (x , y) = centre of mass position and (a, b) = particles position.

Following this and using mR^2, I am getting some odd numbers. for the first particle the results
are: 28 164/169??

thank you again for your help!

 

[Doc Al]

- for the length of each particle to the centre of mass I am using:

d = *squareroot* [ (a-x)^2 + (b - y)^2 ]

where (x , y) = centre of mass position and (a, b) = particles position.

That should work fine.

Following this and using mR^2, I am getting some odd numbers. for the first particle the results
are: 28 164/169??

No, that's not right. Show how you got that result, step by step.

 

[richardnumber]

well for the length of the first particle 'A', i have: sqroot**( (2-11/13)^2 + (0-9/13)^2)**

which results in 3 sqroot*34*/13, is this correct so far? with mR^2, will 'm' represent the mass of each particle? if so, i get 7 41/169, different arithmetic :s thanks again!

 

[ideasrule]

Yes, that's right. Do you really have to work in fractions? This will be much easier if you just use decimals.