# A rod at rest on ice is struck by a piece of clay....

• Truman I
In summary, the problem involves a rod at rest on ice being struck by a piece of clay, which sticks to the rod. The velocity of the center of mass and the angular velocity of the rod after the collision are sought. If the clay does not stick and rebounds with a velocity of 0m/s, the velocity of the center of mass and the angular velocity of the rod are also requested. The problem involves conservation of angular momentum and the moment of inertia of the rod with respect to its axis.
Truman I
1. Homework Statement

A rod (m=6kg, L=3m) at rest on ice (μ=0) is struck by a piece of clay (m=1kg, V=5m/s). The clay sticks.
1) What is the Velocity of the center of mass after the collision?
2) What is the Angular Velocity of the rod following the collision?

If the clay did not stick but its rebound was V=0m/s...
3) Now what is the velocity of the center of mass?
4) And what is the Angular Velocity of the rod?

## Homework Equations

I am just getting back from Spring break, so I'm pretty much brain dead right now. All I know is that I have a test soon and the teacher said a problem like this may be on it. So, this is my attempt to recreate the problem he described. I realize that there are three other threads with similar problems, but I am still having a hard time understanding this concept. Any help would be much appreciated.

What I do understand is that obviously this problem has a lot to do with conservation of angular momentum.

Therefore Li=Lf.

I also understand the L=Iω.

## The Attempt at a Solution

So, taking this, I plugged in what I know.

L=Iω
L=(Irod+Iparticle
L=(1/3 mL2+mr2

→→→→→→→→→→→→→→→→→→→→

From here I am totally stuck; I have no idea how to solve this problem. So, I turned to the AP Physics C: Mechanics test from 2005. One of the Free Response Questions follows this same concept.

It's Question 3

Once again, it's not the same question, so please don't try and help me with this AP one. It's just a similar problem.

All help is much appreciated.

Truman I said:
1. Homework Statement

A rod (m=6kg, L=3m) at rest on ice (μ=0) is struck by a piece of clay (m=1kg, V=5m/s). The clay sticks.
1) What is the Velocity of the center of mass after the collision?
2) What is the Angular Velocity of the rod following the collision?

If the clay did not stick but its rebound was V=0m/s...
3) Now what is the velocity of the center of mass?
4) And what is the Angular Velocity of the rod?

## Homework Equations

I am just getting back from Spring break, so I'm pretty much brain dead right now. All I know is that I have a test soon and the teacher said a problem like this may be on it. So, this is my attempt to recreate the problem he described. I realize that there are three other threads with similar problems, but I am still having a hard time understanding this concept. Any help would be much appreciated.

What I do understand is that obviously this problem has a lot to do with conservation of angular momentum.

Therefore Li=Lf.

I also understand the L=Iω.

## The Attempt at a Solution

So, taking this, I plugged in what I know.

L=Iω
L=(Irod+Iparticle
L=(1/3 mL2+mr2

→→→→→→→→→→→→→→→→→→→→

From here I am totally stuck; I have no idea how to solve this problem. So, I turned to the AP Physics C: Mechanics test from 2005. One of the Free Response Questions follows this same concept.

It's Question 3

Once again, it's not the same question, so please don't try and help me with this AP one. It's just a similar problem.

All help is much appreciated.
Where does the piece of clay hit the rod, at the end?
You said that the angular momentum conserves during the collision. What else?
Specify the symbols you use. What is r?
You used the moment or inertia of the rod with respect to one end. Is it fixed?

ehild said:
Where does the piece of clay hit the rod, at the end?
You said that the angular momentum conserves during the collision. What else?
Specify the symbols you use. What is r?
You used the moment or inertia of the rod with respect to one end. Is it fixed?

Yeah, there really is not much to go on for this problem. The rod is not fixed. The clay hits at one of the ends of the rod. r is meaningless, it was just me assigning a variable out of desperation because I have no idea what I am doing. To be honest, my attempt at the probably is probably fundamentally wrong. I think what I am failing to understand is something more central to the core of the problem.

Is there anything else I need to clear up?

Is the velocity of the center of mass the same before and after the collision? Because then I could just find it with the center of mass formula and use the 5m/s ν0 to solve that part of the problem.

Truman I said:
Is the velocity of the center of mass the same before and after the collision?
If you mean the velocity of the centre of mass of the whole system, rod plus clay, yes.

But there is no need to calculate the system's centre of mass. Just apply conservation of linear momentum and conservation of angular momentum.
For the latter, you need to pick a reference axis and use it consistently. Choose some point that remains fixed. I suggest the point on the ice where the centre of the rod is initially.
What is the initial angular momentum of the system about that point?

I would assume the clay hits the rod at one end, perpendicularly to it.

haruspex said:
If you mean the velocity of the centre of mass of the whole system, rod plus clay, yes.

But there is no need to calculate the system's centre of mass. Just apply conservation of linear momentum and conservation of angular momentum.
Thanks for your response, I'm starting to understand this a bit more.

Yes, I do mean the center of mass of the whole system.

What I don't understand right now is how linear momentum comes into play for this problem. Could you explain that?

haruspex said:
For the latter, you need to pick a reference axis and use it consistently. Choose some point that remains fixed. I suggest the point on the ice where the centre of the rod is initially.
What is the initial angular momentum of the system about that point?

This part makes sense. I can work my way through that. Thanks again.

Truman I said:
What I don't understand right now is how linear momentum comes into play for this problem. Could you explain that?
There are no horizontal external forces, so it will be conserved. That gives a useful equation. Which part don't you understand?

haruspex said:
There are no horizontal external forces, so it will be conserved. That gives a useful equation. Which part don't you understand?
Are you saying like this?

Initial:
Rod: mass=6kg, velocity=0m/s, ρ=0kgm/s
Clay: mass=1kg, velocity=5m/s, ρ=5kgm/s

Final:
Rod and Clay together: mass=7kg, ρ=5kgm/s, so therefore velocity=.714m/s

If that's all I need to do with linear momentum, then I understand that. If there are more steps, or if there is something else I can use linear momentum to solve for in this problem, please let me know.

Truman I said:
Are you saying like this?

Initial:
Rod: mass=6kg, velocity=0m/s, ρ=0kgm/s
Clay: mass=1kg, velocity=5m/s, ρ=5kgm/s

Final:
Rod and Clay together: mass=7kg, ρ=5kgm/s, so therefore velocity=.714m/s

If that's all I need to do with linear momentum, then I understand that. If there are more steps, or if there is something else I can use linear momentum to solve for in this problem, please let me know.
Yes, that's it for the linear momentum (but the usual symbol is p, not ρ).
The angular momentum equation, which you also need, is a little tougher. And you are likely to need a third equation relating linear and angular velocities.

haruspex said:
Yes, that's it for the linear momentum (but the usual symbol is p, not ρ).
The angular momentum equation, which you also need, is a little tougher. And you are likely to need a third equation relating linear and angular velocities.
Thanks for your replies. They have been very insightful. We went over a few more things in class today that I can now connect everything together. I can solve the problem now.

Thanks again.

## 1. What happens to the rod after it is struck by the clay?

The rod will start to rotate due to the impact of the clay. This is because the clay transfers some of its momentum to the rod, causing it to start moving.

## 2. Will the rod continue to move after the impact?

Yes, the rod will continue to rotate due to the conservation of angular momentum. This means that the total angular momentum before and after the impact will be the same.

## 3. What factors can affect the rotation of the rod after the impact?

The rotation of the rod can be affected by the mass and velocity of the clay, as well as the length and mass distribution of the rod. The surface friction of the ice may also play a role in the rotation.

## 4. Will the impact affect the center of mass of the rod?

Yes, the impact of the clay will shift the center of mass of the rod slightly. This is due to the transfer of momentum from the clay to the rod, causing the center of mass to shift in the direction of the impact.

## 5. What is the significance of this experiment?

This experiment demonstrates the conservation of angular momentum and how it applies to objects in motion. It also highlights the importance of understanding the properties of different materials in order to predict the outcome of collisions and interactions between them.

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