Why does torque increase with increasing moment of inertia?

[makamo66]

I'm confused because torque increases with increasing moment of inertia (MOI) but angular velocity decreases with increased MOI because of conservation of angular momentum so angular acceleration would also decrease. And then power is the product of torque and angular velocity so with a smaller angular velocity, there would be less power.

 

[weirdoguy]

I'm confused because torque increases with increasing moment of inertia

How?

because of conservation of angular momentum

And when is angular momentum conserved? Can you tell us what is the scenario you have in mind?

 

[vanhees71]

Obviously this is about the rotation of a body around a fixed axis. If $\Theta$ is the moment of inertia, then the torque is given by
$$\tau=\Theta \dot{\omega},$$
where $\omega$ is the angular velocity.

 

[DaveE]

In order to get a feel for this, I always compared it to linear motion where $F=ma=m \dot{v}$. Of course it's an analogy, so it's ultimately not true, but really similar. MOI is like mass, it makes things harder to spin.

 

[makamo66]

Conservation of angular momentum applies when torque is zero so that must be the clue.

 

[jack action]

torque increases with increasing moment of inertia (MOI) but angular velocity decreases with increased MOI because of conservation of angular momentum

Something is wrong with that statement somewhere. Just the fact that there is a torque $T$ then, by definition, the angular momentum $L$ has changed as well. It wasn't conserved.
$$T = \frac{dL}{dt} = \frac{d(I\omega)}{dt} = I\frac{\partial \omega}{\partial t} + \omega \frac{\partial I}{\partial t}$$

Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque.

 

[Lnewqban]

I'm confused because torque increases with increasing moment of inertia (MOI) but angular velocity decreases with increased MOI...

Please, see these links:
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

http://hyperphysics.phy-astr.gsu.edu/hbase/rstoo.html#sm

Just in case this thread is related to your practice of martial arts, the following video shows how increasing angular velocity is achieved by a sequential use of different muscles, which induce rotation to different parts of the body.

As you can see, during the explained phases, the different moments of inertia are also changed in a sequence for torso (bending), arms (retracting) and leg (extending), in order to transfer impulse to the foot and to achieve the higher speed of impact.

 

[makamo66]

Thank you!

 

[vanhees71]

In order to get a feel for this, I always compared it to linear motion where $F=ma=m \dot{v}$. Of course it's an analogy, so it's ultimately not true, but really similar. MOI is like mass, it makes things harder to spin.

That's true, but only for rotations around a fixed axis you can describe it in this simple way. For general rotations "inertia" has to be described by the tensor of inertia. Then the analogy is quite perfect for the rotation around a fixed point (a spinning top):
$$\vec{L}=\hat{\Theta} \vec{\omega}, \quad \dot{\vec{L}}=\vec{\tau},$$
where $\vec{L}$ is angular momentum, $\hat{\Theta}$ the tensor of inertia, $\vec{\omega}$ angular velocity, and $\vec{\tau}$ torque.

 

[jbriggs444]

Whenever you say that this one thing is proportional to this other thing, you need to pay attention to what is being held constant.

If you are holding angular acceleration constant then torque is directly proportional to moment of inertia.

If you are holding moment of inertia constant then torque is directly proportional to angular acceleration.

If you are holding torque constant then angular acceleration is inversely proportional to moment of inertia.

If you are holding angular momentum constant then angular velocity is inversely proportional to moment of inertia.

 

[makamo66]

I know it was stupid to say I don't understand torque but I was mislead by the following statements:

"The front kick is a straight kick not requiring any turning movements, which makes it faster to execute. Roundhouse and Sidekick both require turning the body sideways but roundhouse is executed quickly due to the concept of Moment of Inertia and Torque. While performing the round house kick, the body is turned while one swings the leg. This increases the distribution of mass around the axis of rotation resulting into a higher moment of inertia and the swinging leg provides a longer lever arm for generating the additional torque."

The statements above make it sound like the angular momentum is being conserved while a torque is being established.

Text above taken from p 30-31
https://www.imse.iastate.edu/files/2014/03/SinghAnupam-thesis.pdf

 

[jbriggs444]

Do not try to learn mechanics from a mixed martial arts presentation. There is a tendency to import scientific sounding terms without much clarity.

By "increasing the distribution" they seem to mean "increasing the [root mean square] average distance of the kicker's mass from their axis of rotation". Yes, this will increase the moment of inertia. It will increase the moment of inertia because moment of inertia is defined as the sum of the bits of the kicker's mass times the squared distance of each bit from the axis of rotation.

The "longer lever arm for generating the additional torque" is word salad. I have no idea what thought it is intended to convey. We want additional impact on the target, not additional reaction torque on the kicker.

 

[Lnewqban]

I believe that the limitation is in the capability of our muscles to contract themselves very strongly and quickly at the same time.
There is also the problem of inertia (both, rotational and linear) of the moving parts of the body.

There is no conserved angular momentum to discuss because the different muscles are giving energy to the leg practically during the full movement.
It feels easy to initiate a quick rotation if your legs and arms are radially closer to the torso.

Velocity of impact is very important for its effectiveness (think of Bruce Lee almost invisible strikes: relatively small mass, lightning speed).
A slower developing circular kick performed with a fully extended leg from beginning to end would be very easy for a defender to figure out, and react to, and defend himself from.

 

[sophiecentaur]

Conservation of angular momentum applies when torque is zero so that must be the clue.

But that's a 'Zero equals zero' which is not very interesting. Usually we consider when torque from one thing is applied to another thing. In that case, the TOTAL angular momentum of the two objects is concerved.