Conservation of Angular Momentum: Hopping onto a Board

In summary, in the given situation of a skater hopping onto a long board sliding on frictionless ice, the conservation of angular and linear momentum can be explained by the absence of outside forces or torques. From a chosen reference point, the sum of angular momentum will be zero, as different points on the board will rotate in opposite directions. The vertical linear momentum is not conserved due to the normal force exerted by the ice, but the mass center of the system will continue to move in a straight line.
  • #1
The Head
144
2

Homework Statement


A long board is free to slide on a sheet of frictionless ice. A skater skates to the board (laid horizontally relative to the skater's motion) and hops onto one end, causing the board to slide an rotate. In this situation, are angular and linear momentum conserved?

Homework Equations


sum of torques = I* alpha
sum of forces = ma

The Attempt at a Solution


I understand that the reason that the angular momentum and linear momentum are conserved is because there are no outside forces or torques, so mathematically it makes sense. I'm struggling with this conceptually though.

So when this board begins to slide and rotate, are we saying the angular momentum is zero? Because if the board were to rotate, so would the person attached to it. I can see that the board will eventually straighten out, but and then just stay straight, but I'm caught up on the part where it's sliding.

Also, I imagine there is friction of some sort to keep the skater on the board (and get the board moving). But in order for the board to slide, wouldn't the board also have to slide against the skater's shoes in order to straighten out? Otherwise, it seems as if the board would just stay horizontal relative to the skater's line of motion. And if there's friction, there would be some loss of energy. I'm less concerned about this at the moment, and more so about the bit in the second paragraph, but it's just not clicking for me fully and I'd like to completely understand.
 
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  • #2
The Head said:
So when this board begins to slide and rotate, are we saying the angular momentum is zero?
Was the angular momentum zero before the collision? (Also note that angular momentum depends on your choice of reference as there is a net linear momentum in the system)

The Head said:
I can see that the board will eventually straighten out, but and then just stay straight, but I'm caught up on the part where it's sliding.
What do you mean by this? If the ice is frictioless the board will rotate indefinitely.
 
  • #3
So if I choose the reference frame of a point on the ice in front of the board where the skater will hit it, the board isn't rotating wrt the point before contact and neither is the skater, so I would think not, but maybe I've got something wrong.
 
  • #4
The Head said:
if I choose the reference frame of a point on the ice in front of the board where the skater will hit it, the board isn't rotating wrt the point before contact and neither is the skater,
Right.
The Head said:
so I would think not
... not what?
 
  • #5
See above. I was answering Orodruin's question about if there is an intial angular momentum. And I thought not.
 
  • #6
The Head said:
See above. I was answering Orodruin's question about if there is an intial angular momentum. And I thought not.
Ok, but what Orodruin actually asked was
Orodruin said:
Was the angular momentum zero before the collision?
So a "no" to that would mean it was nonzero.

Anyway, yes, with that reference axis there is no angular momentum.
 
  • #7
The Head said:
in order for the board to slide, wouldn't the board also have to slide against the skater's shoes in order to straighten out?
No. On landing on the board there is a coalescence, i.e. a completely inelastic collision. Well, it's a bit more complicated than that, but that is the effect.
 
  • #8
So it seems the sum angular momentum is zero about any given point. That is some points of mass will be rotating clockwise, while others will be rotating counterclockwise. Is that a correct way of reconciling this?

Also, from earlier, Orodruin mentioned the rotation would happen indefinitely. So would this skater/board system be making some larger loop around the rink?
 
  • #9
The Head said:
I understand that the reason that the angular momentum and linear momentum are conserved is because there are no outside forces or torques, so mathematically it makes sense. I'm struggling with this conceptually though.
The answer will not be complete without noting that linear momentum in the vertical direction is not conserved because of the normal force exerted by the ice.
 
  • #10
kuruman said:
The answer will not be complete without noting that linear momentum in the vertical direction is not conserved because of the normal force exerted by the ice.
I think we can take the leap onto the board as essentially horizontal. The vertical impulses will all cancel out.
 
  • #11
The Head said:
So it seems the sum angular momentum is zero about any given point.
About any point along the line of the skater's original momentum.
The Head said:
some points of mass will be rotating clockwise, while others will be rotating counterclockwise.
From the point of view of that chosen axis, yes. Imagine standing on the ice facing the skater head on as the leap is made onto the board. Soon after, the skater, and some of the board, will still be approaching you, but a bit to one side of that original line. Meanwhile, the other end of the board will be even further from that line and moving away from you. That is how these angular momenta can cancel.
The Head said:
Orodruin mentioned the rotation would happen indefinitely. So would this skater/board system be making some larger loop around the rink?
Think about what happens to the mass centre. That corresponds to a fixed point on the board, and it will move in a straight line.
 
  • #12
haruspex said:
About any point along the line of the skater's original momentum.

From the point of view of that chosen axis, yes. Imagine standing on the ice facing the skater head on as the leap is made onto the board. Soon after, the skater, and some of the board, will still be approaching you, but a bit to one side of that original line. Meanwhile, the other end of the board will be even further from that line and moving away from you. That is how these angular momenta can cancel.

Think about what happens to the mass centre. That corresponds to a fixed point on the board, and it will move in a straight line.

Great, thank you. That last part really makes sense.
 

1. What is conservation of angular momentum?

The conservation of angular momentum is a fundamental law of physics that states that the total angular momentum of a system remains constant over time, unless acted upon by an external torque. Angular momentum is a measure of the rotational motion of an object.

2. How does hopping onto a board demonstrate conservation of angular momentum?

When a person hops onto a board, they apply a force at a certain distance from the center of mass of the system (the person and the board). This creates an angular momentum in the system. As the person and the board rotate, their individual angular momenta cancel out, resulting in a constant total angular momentum of the system.

3. What factors affect the conservation of angular momentum in this scenario?

The conservation of angular momentum is affected by the mass and velocity of the person, the length and weight distribution of the board, and the distance between the person and the center of mass of the system. Any changes in these factors will result in a change in the total angular momentum of the system.

4. Can conservation of angular momentum be violated?

No, the conservation of angular momentum is a fundamental law of physics and cannot be violated. However, in certain scenarios, it may appear to be violated due to external torques that are not accounted for or if the system is not closed.

5. How is conservation of angular momentum related to other laws of physics?

The conservation of angular momentum is closely related to the conservation of energy and the conservation of linear momentum. These three laws work together to describe the motion of objects in a system and are fundamental principles in physics. Additionally, the conservation of angular momentum is a consequence of Newton's laws of motion and the law of action and reaction.

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