Conservation of Linear and Angular Momentum

[NeuronalMan]

Homework Statement

We have two spheres of equal mass, M, and radius R. One of them is initially at rest (in space, apparently) and the other is moving downwards with a velocity v0. Air resistance is given by an approximation (see below). Consider a completely inelastic, linear collision between the two spheres. What is the velocity v1 of the center of mass of the system of two spheres, after the collision? And similarly, what is the angular velocity omega1 of the sytem?

Homework Equations

The only equation that I think might be relevant is that of air resistance, which is Fv = -kvV, where kv is a constant. This is to say that the air resistance is proportional to the velocity, though. Obviously other relevant equations would be conservation of linear and angular momentum, but these I consider to be so fundamental that I don't write them down.

The Attempt at a Solution

In problems like these, one is inclined to use conservation of momentum, and that is what first crossed my thoughts when I looked at this problem. As for the angular momentum, the net external torque is zero (at least for the linear case), so angular momentum is conserved. That should make it easy to find the angular velocity omega1 for the system of two spheres. However, I have never seen any problem like this, concerning linear momentum. It seems to me that the linear momentum cannot be conserved during the collision, because the net force acting on the system, isn't zero. Namely the weight due to gravity and air resistance. Thus, I am left with no other way of finding the linear velocity v1.

There is also a general case in which the line between the centers of the spheres, form an angle, theta, with the horizontal. And in the general case, the same frustration arises for angular momentum, as I believe air resistance act with a torque on the system.

I didn't yet take the time to learn the LaTeX-codes for writing formulas and equations, but I hope you can bear with my symbols.

 

[alphysicist]

Hi NeuronalMan,

It is true that the net force on the system is not zero if there is gravity and air resistance. However, if the time duration of the collision is very small, we can consider momentum to be conserved, because the momentum relation is (for average external force):

$$\vec F_{\rm ext} \Delta t = \Delta \vec p$$

So if $\Delta t$ is vanishingly small, then momentum is for practical purposes conserved over the time interval of the collision.

 

[Lord Crc]

I'm having an assignment with the exact same problem. In it there's a sentence saying that we can consider the collision to be instantaneous and thus we can ignore the gravity and air resistance.

 

[NeuronalMan]

Hi NeuronalMan,

It is true that the net force on the system is not zero if there is gravity and air resistance. However, if the time duration of the collision is very small, we can consider momentum to be conserved, because the momentum relation is (for average external force):

$$\vec F_{\rm ext} \Delta t = \Delta \vec p$$

So if $\Delta t$ is vanishingly small, then momentum is for practical purposes conserved over the time interval of the collision.

Thank you for your reply--it was most helpful.

 

[NeuronalMan]

I'm having an assignment with the exact same problem. In it there's a sentence saying that we can consider the collision to be instantaneous and thus we can ignore the gravity and air resistance.

To my great disdain, there is no such sentence in my assignment, and as happens, I feel compelled to do things the hard way.

 

[Lord Crc]

The sentence has been added between the 3rd and 4th revision of the assignment... perhaps that's why?

Just find it odd if we don't have the same assignment, considering the striking similarities :)