Ball hits a pivoting rod, what torque changes the angular momentum of the ball?

In summary: $$where the minus sign indicates that torque is applied in the opposite direction to what it was before.
  • #1
alkaspeltzar
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Okay, i know that as a ball collides with a pivoting rod on an axis, the ball has angular momentum. Therefore after the collision, the ball is stopped or slowed, and the rod swings.

The ball provides a force and torque to the rod. But if I isolate the ball, isn't the only thing acting on the ball the reaction force? So the ball slows. But what creates a change in its angular momentum, doesn't that require a torque. And if there is a torque on the ball, why doesn't the ball rotate?
 
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  • #2
alkaspeltzar said:
Summary:: Given a situation where a ball(moving in straight line) collides with a pivoting rod, the ball imparts angular momentum to the rod. How does the ball lose angular momentum?

Okay, i know that as a ball collides with a pivoting rod on an axis, the ball has angular momentum. Therefore after the collision, the ball is stopped or slowed, and the rod swings.

The ball provides a force and torque to the rod. But if I isolate the ball, isn't the only thing acting on the ball the reaction force? So the ball slows. But what creates a change in its angular momentum, doesn't that require a torque. And if there is a torque on the ball, why doesn't the ball rotate?
Angular momentum is defined relative to a point (or axis). The ball has angular momentum about the centre of the rod. In fact, a body moving with constant velocity in a straight line has angular momentum about any point not on that line. Angular momentum is not only about rotation.
 
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  • #3
PeroK said:
Angular momentum is defined relative to a point (or axis). The ball has angular momentum about the centre of the rod. In fact, a body moving with constant velocity in a straight line has angular momentum about any point not on that line. Angular momentum is not only about rotation.
But what reduces the angular momentum of the ball? Doesn't torque cause a change in angular momentum? But there is no torque on the ball
 
  • #4
alkaspeltzar said:
But what reduces the angular momentum of the ball? Doesn't torque cause a change in angular momentum? But there is no torque on the ball
The ball exerts a torque on the rod that changes the rod's angular momentum. According to Newton's third law the rod exerts an equal and opposite torque to the ball that changes the ball's angular momentum.
 
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  • #5
alkaspeltzar said:
But what reduces the angular momentum of the ball? Doesn't torque cause a change in angular momentum? But there is no torque on the ball
Torque, like AM, is always relative to a point. In this case there is an equal and opposite torque on the ball about the centre of the rod. There is no torque about the centre of mass of the ball.

Note that you are still missing the point that the AM of the ball is as a result of its linear momentum; and not rotation about its centre of mass. Angular momentum is not only about rotation.
 
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  • #6
alkaspeltzar said:
Summary:: Given a situation where a ball(moving in straight line) collides with a pivoting rod, the ball imparts angular momentum to the rod. How does the ball lose angular momentum?

But if I isolate the ball, isn't the only thing acting on the ball the reaction force? So the ball slows. But what creates a change in its angular momentum, doesn't that require a torque. And if there is a torque on the ball, why doesn't the ball rotate?
When you speak of angular momentum you always need to specify which axis the angular momentum is measured about.

In this case, the ball starts with no angular momentum about its center of mass, no torque is applied about its center of mass, so it ends with no angular momentum about the center of mass.

Alternatively, the ball has angular momentum about the pivot, a torque is applied about the pivot, the ball loses angular momentum about the pivot
 
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  • #7
Maybe some maths will help. If the ball, of mass ##m## and initially moving at a speed ##v_i## in a direction perpendicular to the rod, collides with the end of the rod of length ##l## which is hinged at the other end, then the initial angular momentum of the entire configuration w.r.t. a coordinate system with its origin ##O## at the hinge is$$\vec{L}_i = \vec{L}_{\text{i, ball}} + \vec{L}_{\text{i, rod}} = m\vec{r}_\text{ball} \times \vec{v}_{\text{i, ball}} + \vec{0} = mlv_i \hat{z}$$where the ##\hat{z}## unit vector is orthogonal to the plane defined by the rod and the ball's velocity. After the collision, the rod gains an angular velocity ##\vec{\omega}## and the ball's speed is reduced to ##v_f##. Because the impulse on the ball is parallel to its initial velocity, it doesn't change direction:$$\vec{L}_f = \vec{L}_{\text{f, ball}} + \vec{L}_{\text{f, rod}} = m\vec{r}_{\text{ball}} \times \vec{v}_\text{f, ball} + I \omega \hat{z} = (mlv_f + I \omega)\hat{z}$$where ##I## is the moment of inertia of the rod about its end. There is zero external torque on the configuration, so ##(\vec{L}_i)_z = (\vec{L}_f)_z##, or more simply ##mlv_i = mlv_f + I \omega##.

Note that, as mentioned above, the rod exerts zero torque about the centre of mass of the ball, so there is no spin component to its total angular momentum ##\vec{L}_{\text{ball}}## about the hinge.
 
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  • #8
kuruman and Perok, i think your responses were helpful and answered my questions.

So to check my understanding, there is a reaction torque from rod based on the axis of rotation, creating a force on the ball, such then the ball slows down and yet relative to the point reduces it angular momentum

Yes the ball sees a force and slows, but relative to the point, we can see the torque and how it keeps AM conserved. IT has to be based on the point of rotation.

Correct?
 
  • #9
alkaspeltzar said:
kuruman and Perok, i think your responses were helpful and answered my questions.

So to check my understanding, there is a reaction torque from rod based on the axis of rotation, creating a force on the ball, such then the ball slows down and yet relative to the point reduces it angular momentum

Yes the ball sees a force and slows, but relative to the point, we can see the torque and how it keeps AM conserved. IT has to be based on the point of rotation.

Correct?
Well, I would say that a force creates a torque about any point not on the line of the force. In this problem it is most convenient to look at torques about the pivot. Why? Because there is also a force at the pivot. If you look at torque and AM about any other point then you must take the force at the pivot into account. This is because that force has zero torque about the pivot but non-zero torque about other points.

Angular momentum about the pivot is conserved because the only external force is at the pivot. Angular momentum about any other point is not conserved - because of the external force at the pivot.
 
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  • #10
PeroK said:
Well, I would say that a force creates a torque about any point not on the line of the force. In this problem it is most convenient to look at torques about the pivot. Why? Because there is also a force at the pivot. If you look at torque and AM about any other point then you must take the force at the pivot into account. This is because that force has zero torque about the pivot but non-zero torque about other points.

Angular momentum about the pivot is conserved because the only external force is at the pivot. Angular momentum about any other point is not conserved - because of the external force at the pivot.

I understand that. I was just trying to simply confirm that the force from the ball on the rod, which creates a torque about the pivot point on the rod, also creates a reaction torque about the same point but now on the ball. Hence the ball slows down and angular momentum is conserved. Simplistically, isn't that correct?
 
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  • #11
alkaspeltzar said:
I understand that. I was just trying to simply confirm that the force from the ball on the rod, which creates a torque about the pivot point on the rod, also creates a reaction torque about the same point but now on the ball. Hence the ball slows down and angular momentum is conserved. Simplistically, isn't that correct?
That's Newton's third law. Angular momentum about the pivot is conserved.

Whenever you write or say "torque" or "angular momentum" the next word must be "about ...". In problems like this at least.
 
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  • #12
PeroK said:
That's Newton's third law. Angular momentum about the pivot is conserved.

Whenever you write or say "torque" or "angular momentum" the next word must be "about ...". In problems like this at least.
Okay that's it right? I got it, can you please confirm? Been on my mind, sorry to ask. Thank you
 
  • #13
alkaspeltzar said:
Okay that's it right? I got it, can you please confirm? Been on my mind, sorry to ask. Thank you
Confirm what?
 
  • #14
I will stick my neck out and confirm that the ball slows down because its angular momentum about the pivot is reduced by the reaction torque about the pivot generated by the reaction force exerted by the rod on the ball.
 
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  • #15
kuruman said:
I will stick my neck out and confirm that the ball slows down because its angular momentum about the pivot is reduced by the reaction torque about the pivot generated by the reaction force exerted by the rod on the ball.
Thank you
 

Related to Ball hits a pivoting rod, what torque changes the angular momentum of the ball?

1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to an object by the distance from the axis of rotation.

2. How is torque related to angular momentum?

Torque is directly related to angular momentum, as it is the force that causes changes in an object's angular momentum. The greater the torque applied to an object, the greater the change in its angular momentum.

3. What is angular momentum?

Angular momentum is a measure of an object's tendency to continue rotating at a constant speed around an axis. It is calculated by multiplying an object's moment of inertia by its angular velocity.

4. How does a pivoting rod affect the torque and angular momentum of a ball?

A pivoting rod acts as an axis of rotation for the ball. When the ball hits the rod, the force of the impact creates a torque on the ball, causing its angular momentum to change.

5. What factors can affect the torque and angular momentum of a ball hitting a pivoting rod?

The torque and angular momentum of a ball hitting a pivoting rod can be affected by the mass and velocity of the ball, the length and orientation of the rod, and the angle and location of the impact on the rod.

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