Question

What is the total moment of inertia of a system of several objects rotating together

Answer 1

Answer:

Lets take

Point mass of objects are m₁ ,m₂ ,m₃ ---and so on

The distance from axis of rotation is r₁ ,r₂ ,r₃ ----and so on  

We know that moment of inertia I  given as

I=∫dm .r²  kg.m²  

I = m₁ r₁² +m₂r₂²+m₃r₃² ------------- kg.m²  

Note-

1 .For point mass

 I=  m r²

2.For disk

$I=\dfrac{mr^2}{2}$

3. For circular ring

 I=  m r²

4. For rod length of L

$I=\dfrac{mL^2}{12}$

If all the objects are rotating together then the total moment of inertia will be summation of the moment of inertia of the all objects

I(total) = I₁ +I₂ +I₃+ ------kg.m²  

Answer 2

The total moment of inertia of a system of several objects can be expressed as : I = I₁ + I₂ + I₃ + -----   Kgm²

Assuming

The point masses of  the objects are : M₁ , M₂, M₃

Distance from axis of rotation are ; r₁, r₂, r₃

Given that :

moment of inertia ( I ) ∫ dm*r²   kg.m² = m₁ r₁²  + m₂r₂² + m₃r₃²

For every type of object Mass is involved

• For a point mass : I = mr²
• For a circular ring : I = mr²
• For a rod : I = mL² / 12
• For a disk : I = mr² / 2

Summing up these moment of inertias since they are rotating together

I( total ) = I₁ + I₂ + I₃ + -----   Kgm²

Hence we can conclude that The total moment of inertia of a system of several objects can be expressed as : I = I₁ + I₂ + I₃ + -----   Kgm².

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