Angular Momentum Vector and Torque Vector

[Teclis]

In studying gyroscopic progression, the angular momentum vector is added to the torque vector. As intuitively these two vectors seem to be qualitatively quite different, how do we know that both vectors are in the same vector field and that they can be manipulated using the rules of vector mechanics? Was this fact determined through experimentation and if so could some one please post a link to a reputable paper on the experiments, or can this fact be proven from more basic principles of mechanics?

 

[Dale]

You cannot add torque and angular momentum. They have different units. Similarly you cannot add force and linear momentum or acceleration and velocity.

 

[Vanadium 50]

In studying gyroscopic progression, the angular momentum vector is added to the torque vector

Where do you see this? Torque and angular momentum have different units so they cannot be added.

 

[A.T.]

How do we know that torque and angular momentum can be added?

You have to integrate torque over time to get the change in angular momentum, which can be added to the original angular momentum:

http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html#rvec6

 

[Teclis]

Where do you see this? Torque and angular momentum have different units so they cannot be added.

I thought torque was equal to the moment of inertia multiplied by the angular acceleration? And angular momentum is moment of inertia multiplied by angular velocity? In gyroscopic progression the fixed points from which the radii are measured is different so it doesn't make sense why we should be able to consider the torque to be the derivative of the angular momentum.

 

[A.T.]

it doesn't make sense why we should be able to consider the torque to be the derivative of the angular momentum.

It's the definition of (net) torque.

 

[Vanadium 50]

so it doesn't make sense why we should be able to consider the torque to be the derivative of the angular momentum.

Did you read what I wrote? This is a completely different statement, and doesn't reflect anything that I said.

 

[Teclis]

It's the definition of (net) torque.

Yes but the torque vectors in the gyroscope are not in the same location. The torque vector from gravity is in the center of the gyroscope as is the angular momentum vector, but the torque vector from the circular motion produced by the angular momentum would have to be on the circumference of the gyroscope not at its center so why are you able to combine to two torque vectors if they are not at the same location?

 

[Dale]

I thought torque was equal to the moment of inertia multiplied by the angular acceleration? ... it doesn't make sense why we should be able to consider the torque to be the derivative of the angular momentum.

This is straightforward to show that these two statements are equivalent for a gyroscope.

Since $\vec L=I\vec \omega$ and since $I$ is constant for a gyroscope we get

$$\vec \tau = I \vec \alpha=I \frac{d \vec \omega}{dt} = \frac{d (I \vec \omega)}{dt}=\frac{d\vec L}{dt}$$

two torque vectors

What two torque vectors. There is the torque from gravity. What other torque is there?

The torque vector from gravity is in the center of the gyroscope

The torque from gravity is a torque about the contact point, not about the center of gravity. The $\vec r$ between the force of gravity and and the center of gravity is zero so it doesn’t produce torque about that point.

 

[A.T.]

...but the torque vector from the circular motion produced by the angular momentum would have to be on the circumference of the gyroscope not at its center...

Not sure what you mean here. Are you confused because ΔL is drawn starting at the tip of the L vector? This is just a geometrical way to visualize vector addition. They are actually both taken around the same reference point.

 

[vanhees71]

This is straightforward to show that these two statements are equivalent for a gyroscope.

Since $\vec L=I\vec \omega$ and since $I$ is constant for a gyroscope we get

$$\vec \tau = I \vec \alpha=I \frac{d \vec \omega}{dt} = \frac{d (I \vec \omega)}{dt}=\frac{d\vec L}{dt}$$What two torque vectors. There is the torque from gravity. What other torque is there?The torque from gravity is a torque about the contact point, not about the center of gravity. The $\vec r$ between the force of gravity and and the center of gravity is zero so it doesn’t produce torque about that point.

You have to be careful! The tensor of inertia $I$ is only constant if you work in the body-fixed (non-inertial rotating) frame of reference. Of course, it's convenient to use the principle axis as the body-fixed Cartesian basis. Then the equation of motion of the spinning top is given by Euler's equations:

https://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)

 

[Teclis]

Not sure what you mean here. Are you confused because ΔL is drawn starting at the tip of the L vector? This is just a geometrical way to visualize vector addition. They are actually both taken around the same reference point.

Wouldn't the centripetal acceleration also be a torque?

 

[Dale]

Wouldn't the centripetal acceleration also be a torque?

No. First, acceleration of any kind is the wrong units for torque. Second, the centripetal force has a moment arm of 0 so it produces no torque.

 

[PeroK]

Yes but the torque vectors in the gyroscope are not in the same location. The torque vector from gravity is in the center of the gyroscope as is the angular momentum vector, but the torque vector from the circular motion produced by the angular momentum would have to be on the circumference of the gyroscope not at its center so why are you able to combine to two torque vectors if they are not at the same location?

Have you ever got something spinning by grabbing both sides and twisting it in opposite directions? This is an example of applying torque at two locations simultaneously to increase the angular velocity more effectively.

It's a fundamental assumption that vector quantities can be added. Whether it's force, torque, electric field etc. You can't prove this, but it's a mathematical rule informed by experiment.

If you don't want to add forces or torque, then you'd need to specify how you think they should combine?

For example, imagine two people spin a large object at the same time. If the total amount of torque is not the sum of the two torques, then what is it? Just the larger torque? Just the smaller torque? The average of the two torques? Something else?

It's the same with forces. If a train has two engines, how do the forces from each engine combine? What's the alternative to the total force being the sum of the two forces? Even if one engine is at the front of the train and one at the rear.

What's your alternative to the addition of torque?

 

[A.T.]

Wouldn't the centripetal acceleration also be a torque?

This makes no sense.

 

[Teclis]

Have you ever got something spinning by grabbing both sides and twisting it in opposite directions? This is an example of applying torque at two locations simultaneously to increase the angular velocity more effectively.

It's a fundamental assumption that vector quantities can be added. Whether it's force, torque, electric field etc. You can't prove this, but it's a mathematical rule informed by experiment.

If you don't want to add forces or torque, then you'd need to specify how you think they should combine?

For example, imagine two people spin a large object at the same time. If the total amount of torque is not the sum of the two torques, then what is it? Just the larger torque? Just the smaller torque? The average of the two torques? Something else?

It's the same with forces. If a train has two engines, how do the forces from each engine combine? What's the alternative to the total force being the sum of the two forces? Even if one engine is at the front of the train and one at the rear.

What's your alternative to the addition of torque?

Hi, Thank you very one for taking the time to answer my questions and explain things to me. I think I found the answer I am looking for in video number 14 from the following web site:

http://www.gyroscopes.org/1974lecture.asp

The presenter states that moment inertia and angular momentum are separate properties of an object and can not be derived from Newtons laws of motion. They are separate laws of mechanics formulated form experimental observation and not derived from other laws of physics.

 

[PeroK]

Hi, Thank you very one for taking the time to answer my questions and explain things to me. I think I found the answer I am looking for in video number 14 from the following web site:

http://www.gyroscopes.org/1974lecture.asp

The presenter states that moment inertia and angular momentum are separate properties of an object and can not be derived from Newtons laws of motion. They are separate laws of mechanics formulated form experimental observation and not derived from other laws of physics.

Yes, that's a load of nonsense. The engineer/scientist in question eventually retracted these statements. See here:

https://en.wikipedia.org/wiki/Eric_Laithwaite

 

[A.T.]

The presenter states that moment inertia and angular momentum are separate properties of an object and can not be derived from Newtons laws of motion.

This is wrong. Moment inertia and angular momentum are derived via the sum (integral) over the infinitesimal point masses based on the Newtons laws of motion.

You can explain the behavior of a gyroscope based on linear dynamics, without any mention of torque and angular momentum:

 

[Vanadium 50]

Wouldn't the centripetal acceleration also be a torque?

Maybe it would be a good idea to take a step back. Do you understand vector algebra? What about linear motion? What do you think about the statement "wouldn't the acceleration also be a force?"

 

[vanhees71]

Hi, Thank you very one for taking the time to answer my questions and explain things to me. I think I found the answer I am looking for in video number 14 from the following web site:

http://www.gyroscopes.org/1974lecture.asp

The presenter states that moment inertia and angular momentum are separate properties of an object and can not be derived from Newtons laws of motion. They are separate laws of mechanics formulated form experimental observation and not derived from other laws of physics.

I don't know, what lecture this is, but it's wrong. Of course the laws of the spinning top are derived from Newton's mechanics, using the model of a rigid body.

From the rigidity condition it's derived that a rigid body has 6 degrees of freedom, which is also very intuitive. You start with the description of the body at some initial time in an inertial frame of reference and think about which parameters you need to characterize any point of the body. Obviously all you need is a fixed point within the body (often a good choice for this point is the center of mass) and a Cartesian basis fixed with the body at this point. Now obviously any point of the body is described by a body-fixed position vector which has time-independent components wrt. the inertial basis. Now the motion relative to the inertial reference frame is described by the components of the body-fixed origin of the body-fixed reference frame wrt. the origin of the inertial frame and wrt. the inertial Cartesian basis (three translative degrees of freedom) and the rotation of the body-fixed basis wrt. the inertial basis (three rotational degrees of freedom, e.g., the usual Euler angles).

Then you define the tensor of inertia in the body-fixed reference frame and use the transformation from the vector components in the body-fixed frame to the ones in the inertial frame to express Newton's equation of motion in terms of the body-fixed components. The result is Euler's equation of the spinning top as rerferenced above already, augmented by the equation of motion for the body-reference point relative to the inertial reference frame.

 

[PeroK]

I don't know, what lecture this is, but it's wrong.

See the link in post #7. Laithwaite was quite a character!

 

[A.T.]

See the link in post #7. Laithwaite was quite a character!

Here is a good explanation of Laithwaite's misconception:

 

[Teclis]

Maybe it would be a good idea to take a step back. Do you understand vector algebra? What about linear motion? What do you think about the statement "wouldn't the acceleration also be a force?"

Yes I understand vector algebra I arrived at this topic from working on question 41 of chapter 13.5 of Marsden and Weinstein Calculus III.

I understand linear motion so I would answer that acceleration is the ratio of Force to Mass.

 

[Vanadium 50]

Yes I understand vector algebra

Good. So do you also understand that when you add two vectors they have to have the same units?

 

[Teclis]

Good. So do you also understand that when you add two vectors they have to have the same units?

Yes I understand this.