Why is angular momentum conserved here?

In summary, when the ball is rolled without slipping on a horizontal rough surface, its angular velocity and center of mass are constant vectors.
  • #1
cpgp
14
1
A cylinder of radius R spins with angular velocity w_0 . When the cylinder is gently laid on a plane, it skids for a short time and eventually rolls without slipping. What is the final angular velocity, w_f?

The solution follows from angular momentum conservation. $$L_i = I \omega_0 = L_f = mv_fr + I\omega_f$$
My question is, why is angular momentum conserved? Friction exerts a torque on the ball, so shouldn't its angular momentum be changing with time?
 
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  • #2
cpgp said:
A cylinder of radius R spins with angular velocity w_0 . When the cylinder is gently laid on a plane, it skids for a short time and eventually rolls without slipping. What is the final angular velocity, w_f?

The solution follows from angular momentum conservation. $$L_i = I \omega_0 = L_f = mv_fr + I\omega_f$$
My question is, why is angular momentum conserved? Friction exerts a torque on the ball, so shouldn't its angular momentum be changing with time?
AM is always calculated about a point. In this case, the AM in that equation is taken about a point on the surface. As the force acts in the direction of the surface, the AM about that point is conserved.

You can also solve this problem using AM about the initial position of the centre of the cylinder. In that case, the final AM is different from the initial AM because of the external torque about that point.

In general, if you have an external force, then the AM about a point on the line of that force is conserved.
 
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  • #3
Thanks, I understand now.
 
  • #4
cpgp said:
Thanks, I understand now.
When I first did this problem I used the second method (AM about the centre of the cylinder). Then I saw the solution using conservation of AM about the surface and I thought it was very clever!
 
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  • #5
I remember a nice fact in this regard. There is a horizontal rough enough table. There is a paper sheet on the table.
Then you launch a homogeneous ball to roll on the table. The ball rolls without slipping, its angular velocity ##\boldsymbol \omega## and velocity of its center of mass ##\boldsymbol v## are the constant vectors.

When the ball rolls on the paper sheet you begin to jerk the paper as you wish in any horizontal direction. The ball may slide. When the ball leaves the paper it once stops to slide and rolls without slipping again.

It hard to believe but its angular velocity and the velocity of its center of mass restore to the initial vectors ##\boldsymbol \omega,\quad## ##\boldsymbol v## respectively.
 
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Related to Why is angular momentum conserved here?

1. What is angular momentum and why is it important?

Angular momentum is a physical quantity that describes the rotational motion of an object. It is important because it helps us understand the behavior of rotating objects and is conserved in many physical systems.

2. How is angular momentum conserved?

Angular momentum is conserved when there is no external torque acting on a system. This means that the total angular momentum of a system remains constant, even if individual components of the system change their angular momentum.

3. Why is angular momentum conserved in this specific system?

Angular momentum is conserved in this specific system because there is no external torque acting on it. This could be due to the absence of external forces, or the forces acting on the system are balanced, resulting in a net torque of zero.

4. How does conservation of angular momentum affect the motion of objects?

Conservation of angular momentum affects the motion of objects by causing them to rotate at a constant speed or change their rotational axis. This can be seen in phenomena such as a spinning top maintaining its rotation or a figure skater spinning faster when they pull their arms in.

5. Can angular momentum ever be lost?

No, angular momentum cannot be lost in a closed system. It can only be transferred from one object to another or converted into other forms of energy, such as heat or sound. This is known as the conservation of angular momentum law.

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