hiya

Moment of inertia is to rotation what mass is to linear motion. It measures object's resistance to rotation

J=moment of inertia

a=angular acceleration

J = m * r^2

Torque = m * r^2 * a = J * a


Why is J defined as J = m * r^2? Instead why not

J = m * r => Torque = J * r * a

?
There has to be a reason why that way and not the other. I'm shure it's not just for the sake of convenience!

Also, according to formula J=m*r^2 bigger the radius r is the greater is resistence to rotational acceleration. I would think the opposite would be true since less force is needed to rotate a body if radius is bigger?

thank you

Hi Boris, you may want to look at the corresponding HyperPhysics site: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi

One problem with using J = m r would be for distributed objects where there are multiple points with multiple r's. So which r would you use?

-Dale

hiya Dale

I still don't get it.Sorry

I'm not familiar with the term "distributed". Don't most objects have multiple points each point having its own distance(r) to rotational axis?

I don't see what you're getting at

<i>More</i> force is needed to rotate a body (to the same rotational rate) if the radius is bigger. Ice skaters use this fact all the time: They start rotating with their arms extended, building up a good amount of angular momentum. When they fold their arms in, their rotational inertia decreases and thus their spin rate increases due to conservation of angular momentum.

To answer your main question, the formula for rotational inertia is not a matter of convenience. It is a direct consequence of Newton's Second Law, <i>F=ma</i>.

It seems there are quite a few things I don't understand about moment of inertia and torques in general, so one at the time I guess and hopefully someone will be able to explain it

*torque is needed to rotate a body around the axis. We learned that greater the given distance from the center of rotation, less force we need to exert to get same torque and as a result when lever arm is longer we are able to rotate objects we couldn't with reduced lever arm.

*But with moment of inertia we learn that objects also have tendency to resists a change in the rotational speed of an object. And the longer the lever arm is, greater is resistance of an object to change its rotational speed, so more torque is needed.

Here it is saying that it is much harder to change a rotational speed of an object if lever arm is long.

In short, if we want to rotate a body we will need greater torque when lever arm is longer. And greater torque means we will need to exert more force. Which contradicts the definition of torque, that claims we need to exert less force to rotate an object if lever arm is longer.

So if I plan to rotate heavy object around axis:

-One one hand I will get same torque but I will need to exert much less force if lever arm is longer
-but on the other hand with lever arm longer I will need much greater torque and thus much greater force to start rotating a body, since body will want to stay at rest

This is screaming with contradictions

You're confusing the two concepts. You articulate them both correctly indvidually at the top of your last post - but are just not thinking of them properly.

Remember: Torque = force x radius.

If something is spun about an axis by applying 100N of force at a radius of 1 meter, at a radius of 10 meters there is effectively only a 10N force acting to accelerate the mass at that point. The farther the mass is away from the axis of rotation, the harder it will be to accelerate with a given amount of torque applied.

The figure skating example given above is a very good one.

Conversely, if you're trying to create a certain amount of torque, you can do it by applying a smaller force if the radius is longer. Any simple lever is a good example of this.

I guess I must be missing something very obvious. For now I don't know how and what exactly to ask, but this torque and inertia stuff really got me confused

Why do we need more torque to accelerate mass if radius is longer?

Isn't it then easier to rotate an object when radius is smaller(true more force is needed for same torque but on the other hand we can accelerate it with less torque) ?

Consider a handle which is very light compared to the object. You can make the handle any length you want without changing the total inertia. Then the two concepts are independent: the bigger and heavier the object, the greater its rotational inertia and the more difficult it is to rotate it; the longer the handle, the more leverage you have (more torque for the same force) and the easier it is to move it.

And why is resistance of an object to change its rotational speed greater if lever arm is longer? Why don't both torque and rotational inertia either increase or decrease when lever arm is longer(or shorter), instead of them having opposite effects ?

It depends on the mass of the handle. If it's very light (m is small) it won't make much difference to the overall J.
Remember, J is the sum of m * r^2 for all particles in the object.

Boris,
I think your problem with your understanding of the concepts of rotational behavior is that you are failing to distinguish that rotational inertia, torque, and angular acceleration are three different but related concepts. The linear analogs of these concepts are mass, force, and linear acceleration.

• Mass and rotational inertia are intrinsic characteristics of a body.
• Force and torque are external agents that act on a body.
• Linear and rotational acceleration are a body's responses to forces and torques applied to the body.

To answer some your questions,

• Why is resistance of an object to change its rotational speed greater if lever arm is longer?
  Answer: It isn't. The "resistance" remains constant: rotational inertia is an intrinsic characteristic. Applying the same force at different lever arms will change the angular acceleration, but that is because changing the lever arm changes the torque, not the rotational inertia.
• Why don't both torque and rotational inertia either increase or decrease when lever arm is longer(or shorter), instead of them having opposite effects?
  Answer: Changing the lever arm does indeed change the torque, but it has nothing to do with rotational inertia. Does changing the force applied to an object change the object's mass? No. Force and torque are measures of external actions applied to an object; mass and rotational inertia are intrinisic characteristics of the object.

I think I get it now.

I know it's actually much more complicated than this, but is formula( call it pseudo formula since there isn't really one, or at least it's much different than mine) for normal inertia ( for linear motion ) inertia = 1 * m, meaning it solely depends on mass and as such we know that for object to accelerate with a, it always depends on mass and so F = m * a is in a way F=inertia*a =1*m*a, while rotational inertia besides mass also depends on distance from axis,so : rotational_inertia = m * r^2 ?

You're getting the concept. The distribution of mass is factor in rotational inertia (i.e. how much mass is located where within the object). That's what D H meant as an "intrinsic characteristic" of the body.

When the figure skater is twirling with his arms outstretched he has a greater amount of mass further away from the centre of rotation (so he has a greater rotational inertia) and he spins slowly. When he pulls his arms in, he decreases his rotational inertia and even though he hasn't added any more energy to his spin his rotation speeds up.

How the mass is distributed within the body is not important to a "straight line" translational acceleration as you are thinking of in your last post.