Angular Momentum Term Equals Zero?

In summary, the conversation is discussing a classical mechanics problem involving a table with a hole and two masses on a string. The question is about a certain term being set to zero in the cross product of R x mV, which represents the angular momentum vector K. The conversation includes a derivation of the equation for angular momentum in polar coordinates and a discussion of a possible error in the derivation. The conclusion is that there is an error in the derivation and a correction is suggested.
  • #1
KleZMeR
127
1
Hi All,

This is from a classical mechanics problem, and I already 'solved' the problem, but I'm interested in why a certain term is set to zero. I think I understand the concept but just want to clarify.

The problem is a table with a hole in it and two masses on a string, one mass is hanging through the hole with only a Z component, and the other is on the table with an X and Y component (Z plane).


When I take the cross product of R x mV, I get the angular momentum vector K which has only a 'vertical' component:
R x mV = [m*(r^2)*dθ + m*r*dr*sin(2θ)] K


But I am told that:
R x mV = m*(r^2)*dθ K


The sin(2θ) came from some trig identity work. So I am wondering is this because there is no effect on the K vector from a sin(2θ) factor which is only in the Z plane? Why is this term 0? Is dr = 0 ? I think r is fixed but the problem does say that gravity affects the hanging mass, so perhaps dr in the Z plane is not zero? Any help understanding this is appreciated.
 
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  • #2
There is an error in your derivation of equation for angular momentum. Without seeing the derivation, I cannot say what this error is. A simple, if somewhat cumbersome/ way to obtain the angular momentum in polar coordinates is by writing ## x = r \cos \theta, \ y = r \sin \theta ##, then writing ## \dot x = ..., \ \dot y = ... ## and taking their cross product.
 
  • #3
Here is my attempt, I uploaded it
 

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  • #4
So you have ## mr [\dot r \cos \theta \sin \theta + ... - \dot r \sin \theta \cos \theta + ... ] ## yet you write ## = mr [ ... + 2\dot r \cos \theta \sin \theta] ##.
 
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  • #5


I can provide an explanation for why the angular momentum term is equal to zero in this problem. Angular momentum is a vector quantity that describes the rotational motion of an object. It is calculated as the cross product of the position vector (R) and the linear momentum vector (mV). In this case, the R vector is pointing in the Z direction and the mV vector is in the X-Y plane. When we take the cross product, we get a vector that is perpendicular to both R and mV, which in this case is the K vector in the Z direction.

Now, the sin(2θ) term in the cross product represents the angle between R and mV. However, in this problem, the mV vector is only in the X-Y plane and does not have any Z component. This means that the angle between R and mV is 90 degrees, making the sin(2θ) term equal to zero. This is because the cross product of two vectors that are perpendicular to each other is always zero.

Additionally, the dr term represents the change in the position vector, which in this case is fixed for the hanging mass. This means that dr is equal to zero, making the entire term (dr*sin(2θ)) equal to zero.

In conclusion, the angular momentum term is equal to zero because the cross product of the position vector and linear momentum vector is perpendicular to the plane of motion and there is no change in the position vector for the hanging mass. I hope this helps clarify the concept for you.
 

1. What is Angular Momentum Term Equals Zero?

Angular momentum term equals zero refers to the mathematical equation used to calculate the total angular momentum of a system. It states that the sum of the individual angular momentums of all the components in a system is equal to zero.

2. How is Angular Momentum Term Equals Zero calculated?

The calculation of Angular Momentum Term Equals Zero involves multiplying the angular velocity (ω) of a rotating object by its moment of inertia (I) and the distance (r) from the center of rotation. This is represented by the equation L = Iωr. The sum of all these individual angular momentums is then equal to zero.

3. What is the significance of Angular Momentum Term Equals Zero?

The significance of Angular Momentum Term Equals Zero lies in its application in understanding the behavior of rotating systems. It helps to determine the stability and motion of objects, such as planets, satellites, and even subatomic particles.

4. Can Angular Momentum Term Equals Zero ever be non-zero?

No, according to the law of conservation of angular momentum, the total angular momentum of a system cannot change unless acted upon by an external torque. Therefore, the sum of angular momentums always equals zero in a closed system.

5. How does Angular Momentum Term Equals Zero relate to Newton's First Law of Motion?

Angular Momentum Term Equals Zero is related to Newton's First Law of Motion, also known as the law of inertia. This law states that an object at rest will remain at rest, and an object in motion will continue in a straight line at a constant speed unless acted upon by an external force. The conservation of angular momentum, represented by Angular Momentum Term Equals Zero, is a manifestation of this law in rotational motion.

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