Moment of inertia (compound object)

In summary, the moment of inertia is a property of rotating objects that depends on their mass distribution and the axis of rotation. For simple shapes like a disk or ring, it can be calculated using specific formulas. When combining two objects with different moments of inertia, their individual moments can be added together to find the combined moment of inertia. Torque is the rotational equivalent of force and is dependent on the distance from the axis and the angle of the force. It is often represented as a cross product.
  • #1
mbrmbrg
496
2
I have a disk parallel to the floor rotating about an axis perpindicular to the floor, going through the center of the disk.
Its moment of inertia is [tex]I=\frac{1}{2}mR^2[/tex]
I also have a ring (which I will forbear to describe. A picture is worth a thousand words [or 30-ish, in this case], so please see attatchment) with intertia [tex]I=\frac{1}{2}m(R_1^2+R_2^2)[/tex]
All fine and dandy. But when I place one on top of the other (still rotating about the same axis), to find the moment of inertia, do I just add their individual moments? Or something else?
 

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  • #2
It can easily be seen from the definition of moment of inertia that if you know the moment of inertia of two objects rotating about the same axis the combined moment of inertia is the sum of the two individual moments of inertia.
 
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  • #3
Thank you!
I confess--I am deplorably shaky on the definition of the moment of inertia, and all of torque in general.
 
  • #4
The reason moments of inertia are just given in formulas is because the real definition involves nasty summations and integrals. Take every particle and multiply by the distance from axis squared. For a hoop, its obviously MR^2, but there's a bunch of calculus behind the other formulas, so be thankful for them.

Torque is the rotational equivalent of force. It also depends on the distance from the axis. The longer the arm, the more leverage and the more torque. It is defined as the perpendicular distance, so you need right angles. If a force acts at an angle, the torque is given as rFsintheta, because the sin of the angle yields the perpendicular force. This is also known as a cross product. I hope this helps
 

Related to Moment of inertia (compound object)

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in rotational motion. It is the sum of the products of each individual mass element of the object multiplied by the square of its distance from the axis of rotation.

How is moment of inertia calculated for a compound object?

The moment of inertia for a compound object can be calculated by dividing the object into smaller, simpler shapes and using the parallel axis theorem. This involves calculating the moment of inertia for each individual shape and then adding them together.

What factors affect the moment of inertia of a compound object?

The moment of inertia of a compound object is affected by the distribution of mass, the shape of the object, and the axis of rotation. Objects with a larger mass or mass that is further from the axis of rotation will have a larger moment of inertia.

Why is moment of inertia important in physics?

Moment of inertia is an important concept in physics because it helps us understand and predict an object's behavior when rotating. It is also used in calculations involving torque and angular momentum.

How does moment of inertia differ from mass?

Moment of inertia and mass are related but different concepts. Mass is a measure of an object's resistance to changes in linear motion, while moment of inertia is a measure of an object's resistance to changes in rotational motion. Objects with a larger mass will generally have a larger moment of inertia, but this is not always the case.

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