# Find axis to minimize moment of inertia

• vu10758
In summary, the problem involves two masses connected by a thin rod, and the task is to determine the location of an axis perpendicular to the rod which minimizes the moment of inertia of the system. The solution involves using the formula I = Mx^2 and setting it equal to I_cm + Mh^2. By differentiating and setting the derivative equal to zero, the value of x can be determined. The second part of the problem involves verifying this solution by using the formula I = I_cm + mh^2 and plugging in the values for M1, M2, and L. The result confirms that the minimum moment of inertia is at the axis through the center of mass.
vu10758
Two masses (M1 and M2) are connected by a thin massless rod of length L.

a) Determine the location of an axis perpendicular to the rod) which minimizes the moment of inertia of the system.

The answer is "with M1 at the origin x = M2*L/(M1+M2)

I know that I = Sum of ( M1*r1^2)
However, how do we know that M1 is at origin? I know that M1 + M2 is the total mass of the system. I know L is the length between the two points. But I don't know why the answer is above. I think I am missing something in understanding this but I don't know what.

b) Verify that the moment of inertia about this axis can be obtained by determine I about an axis through M2 and then using the parallel-axis theorem.

I know that I = I_cm + Mh^2 , where h is the distance between the axis, M is the total mass (M1+M2). Is the distance between the axis L? How do I use this to verify the moment of inertia.

for part a)
Hint --
How would you calculate the inertia of the system if the system was rotated about some known position -- say a distance "x" down the rod from the mass M1?

Wouldn't I use the formula I = Mx^2 then where x is the distance down from the rod?

Not quite -- what do you mean by "M"... are both masses contributing?

Ok there's two ways you can do this. First is intuitively. The moment of intertia is effectively a measure of mass for rotational motion. So if you have your axis straight through the middle of the rod, you can picture just through everyday experience that its going to require a fair bit of effort or torque in order to spin it. If you have the rod through one of the masses, the torque required to spin that object is virtually nil (t = fd and if d is some infintesimle radius...) And the only major force drain is the other mass at the max length. As I am sure you'll see from the answer you have there, if the masses are equal its best to have it in the middle, if ones larger its best to have the axis closer to that one, or through it.

This can also be solved mathematically. You have written one definition of I but there's another, more useful (for me anyway) term.

$$I = \int r^2dm$$
Now this more general form is more for complex objects and so your sumation one is actually better for this point particle problem, but its a good idea to keep the above one in mind. Your moment of inertia is going to be

$$I = r^2M1 + (L-r)^2M2$$

right? From your calculus classes you know that if you want to minimize a function you can differentiate it and equate to zero. Bobs your uncle :)

-G

For part 2,

I know that I = I_cm + mh^2, where h is the distance between the axes. So is h, L-r in this case? How can I get I about an axis through M2.

Is it

I = M2*L^2 + M1*(L-r)^2 ?

It is my recollection that the min. I is that axis which is the same as the centor of mass. Am I right?

## 1. What is the moment of inertia?

The moment of inertia is a physical property that measures an object's resistance to rotational motion. It is a measure of how spread out the mass of an object is in relation to its axis of rotation.

## 2. Why is it important to minimize the moment of inertia?

Minimizing the moment of inertia is important because it allows for easier rotation or movement of an object. Objects with lower moments of inertia require less force to rotate, making them easier to handle and control.

## 3. How can I find the axis that minimizes the moment of inertia?

The axis that minimizes the moment of inertia can be found by using the parallel axis theorem. This involves shifting the axis of rotation to a different point on the object and calculating the moment of inertia at that new axis. The axis that gives the lowest moment of inertia is the one that minimizes it.

## 4. What factors affect the moment of inertia?

The moment of inertia is affected by the mass and distribution of mass in an object. Objects with more mass and/or mass concentrated farther from the axis of rotation will have higher moments of inertia.

## 5. How is the moment of inertia used in real-world applications?

The moment of inertia is used in many real-world applications, such as designing rotating machinery, calculating the stability of structures, and predicting the motion of objects in sports or other physical activities. It is also an important concept in understanding and analyzing the behavior of objects in motion, such as in the study of physics and engineering.

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