Elastic Collision of Rod and Cube: Dynamics Unveiled

[arildno]

Consider a cube of uniform density, mass M, sidelengths 2a resting on a frictionless plane.
Origin is placed in the cube's center.

A rod of length L, attached to the ceiling z=a+L, mass m, hits with its tip the corner (-a,a,a) on the side x=-a with velocity V_{0}\vec{i}.

Determine the state of motion after elastic collision(s).

To give a hint:
Neither angular nor linear momenta are conserved..

 

[narblecinder]

trick question! the rod is moving too slowly and is too light to effect the cube by any meaningful amount!

just kidding. but the cube does spin a bit, right?

 

[NoTime]

Collision?
It seems just as fair to say the rod is sitting on the cube as it does to say it was attached to the ceiling.
V=0
So conservation is not an issue.

 

[arildno]

The rod may rotate about its pivot point.
And the cube will start sliding, plus rotating about a couple of axes
(the \vec{k},\vec{j} axes, actually).

 

[NoTime]

The rod may rotate about its pivot point.
And the cube will start sliding, plus rotating about a couple of axes
(the \vec{k},\vec{j} axes, actually).

The rod might be in a highly unstable position, but barring any external forces why should it topple rather than just stand on the corner of the cube

 

[Gokul43201]

The recoil of the rod is not restricted to \hat{i}, since it's hitting a corner ?

This is a toughie...

 

[arildno]

The recoil of the rod is not restricted to \hat{i}, since it's hitting a corner ?

This is a toughie...

Not THAT evil, it undergoes only uniaxial rotation..
Consider the impulse between the rod&cube to be in the \vec{i} direction.

 

[arildno]

The rod might be in a highly unstable position, but barring any external forces why should it topple rather than just stand on the corner of the cube

Consider it to hit the cube at the side x=-a at a point only slightly (insignificantly) displaced from the corner.

 

[NoTime]

Consider it to hit the cube at the side x=-a at a point only slightly (insignificantly) displaced from the corner.

Because it wasn't stated in the original problem
Also a corner would be at sqrt(a^2 + a^2).
Somewhat more than a.

 

[arildno]

Because it wasn't stated in the original problem
Also a corner would be at sqrt(a^2 + a^2).

Nope. the distance from the center to the corner is \sqrt{3}a its (vector) position is (-a,a,a), measured from the origin (i.e, the cube's center).

 

[NoTime]

Nope. the distance from the center to the corner is \sqrt{3}a its (vector) position is (-a,a,a), measured from the origin (i.e, the cube's center).

Good point.
Still don't see why it should fall over.

 

[Gokul43201]

Good point.
Still don't see why it should fall over.

There's no reason to assume it won't. After all the impulse does provide a torge about \hat {j}.

 

[NoTime]

There's no reason to assume it won't. After all the impulse does provide a torge about \hat {j}.

Well, z=a+L and was not given as z=a+L+d.
So being attached or not to the ceiling doesn't seem like a particularly useful bit of information. Although there are certainly considerations that would make it otherwise, none of these seem to be specified. I see no reason to assume them.
My main question from the start is -> What impulse?
Or why isn't it just as correct to say the rod is sitting on the cube?
What reason is there to believe that this situation should change?
Perhaps I am just being overly picky, but it is a brain teaser

Suppose the rod was made of lead with a diameter of 5a with the cube being soft butter. Rotation or sliding seem unlikely in this case.

 

[Gokul43201]

The rod is not sitting on the cube. I think you've misunderstood the problem.

There's this cubical box on the floor. On the ceiling, exactly above one of the corners of the box is a rod hanging from a pivot, such that the bottom end of the rod just touches a top corner of the box. The rod is pulled back along the XZ plane and released. The tip of the rod strikes the YZ face of the box very near this corner, at a velocity v_0.

 

[arildno]

Suppose the rod was made of lead with a diameter of 5a with the cube being soft butter. Rotation or sliding seem unlikely in this case.

A very good objection!
The cube and the rod are to be modeled as ideal rigid bodies; the rod has only length (which was the only dimension I gave), and undergoes constrained motion about its pivot point in the ceiling.
That is, at all times, it is to be considered as a physical pendulum, with an associated angular velocity \omega\vec{j}
hence, initially, \omega_{0}=-\frac{V_{0}}{L}
since we then have:
\omega_{0}\vec{j}\times{(-L\vec{k})}=V_{0}\vec{i}

 

[NoTime]

Ok, the picture is now radically different.
No mention of pivots, rigid or perfectly elastic in the original statement.

Is there a diagram for this setup?

Are there other forces acting on the cube, like gravity?
Or restating, is there a constraint that one face of the cube will remain parallel to the plane?

 

[Gokul43201]

Yes there's gravity. No, there's no other constraint.

 

[arildno]

The "constraints" are those which gives us a reasonable model for analyzing a "real-life" collision.
First, most collisons we see between practically rigid bodies takes place over an extremely short period of time.
How can we maximally utilize the insight of an extremely short collision period to develop a good, simple model of collisions?
Secondly, it is physically unreasonable that the cube is going to enter the table/plane and get stuck deep within it.
How must we take account of this, does it for example mean that we necessarily have to assume that the table/plane imparts an impulse on the cube (and if so, what can we say of that impulse on basis of the information that the plane is frictionless)?

My basic reason for posting this problem, is that I would think someone might be interested in how to extend our ideas/problem-solving skills beyond the treatment of simplistic 2-D collisions ordinarily taught (in which, for the most part, we make do with conservation laws).

 

[NoTime]

Interesting ideas.
It seems to me that if the cube was free floating in a 0g environment and struck on the corner of the x face that it would rotate around the z and y-axis with some drift.

Just a thought.
The surface might elastically deform enough to form a step that would latch the edge.

 

[arildno]

Let's see how the idea of extremely short duration of collision may be used to our advantage concerning external forces of the gravity and air-resistance kind:
Clearly, the magnitude of these forces will be roughly proportional to the duration interval, i.e, we may reasonably expect them to be negligible in comparison to the huge collision forces associated with the actual deformations.

Hence, to obtain a reasonably accurate&simple model of collisions, we should be justified in neglecting the impulses from these forces.
Agreed?

 

[NoTime]

Hence, to obtain a reasonably accurate&simple model of collisions, we should be justified in neglecting the impulses from these forces.
Agreed?

I think it boils down to -> can you construct an experimental system to reliably demonstrate the results of any calculations you might make.

Just a suspicion on my part, but I think what you are proposing would exist if it were not subject to a whole bunch of gotyas.
It's just to cool otherwise

All in all, if you change the frame of reference, your cube and pendulum bear a remarkable resemblance to dice.
The quintessential chaotic system.

 

[arildno]

First off, I do NOT propose to predict the entire motion of the system generated by the collision.
I only want to make predictions for how the velocities have changed JUST AFTER the collision phase; how the non-colliding objects (in particular, the cube) thereafter choose to move, is not my concern.
Hence, the model for the cube should provide the INITIAL conditions for its motion after the collision phase, not the differential equations governing that period.

Secondly, this is certainly not my own invention; it is called classical impact theory, and was one of the first great successes of Newtonian mechanics.
The fact that only the simplest portions of this theory is generally taught today, is not because the theory is inaccurate for 3-D collisions in general, but because the focus of physics today is on completely different phenomena than it was earlier, as it should be.
However, when we, on occasion, DO want to analyze general collision phenomena , unless we want to try and solve some truly ugly differential equations, the only viable "theoretical" approach remains classical impact theory.
An alternate (and IMO, highly important) strategy for analysis, is, of course, the experimentalist approach.

Thirdly, as to chaotic behaviour:
Although it is quite true that several systems are extremely sensitive to initial conditions, and hence, that only the slightest deviation from the "true values" create radically different patterns of motion, this does not mean that EVERY imaginable system experience such effects.
To have a simple, general model to give good estimates of initial values (after a collision) remains quite useful.
Although it might be objected that the particular choice I made for an illustrative example is, for some parameter combinations, extremely chaotic in its ensuing motion, that example was chosen because the objects involved have relatively simple geometries.

I'll proceed further in a while..

 

[ceptimus]

Won't you run into the classic, 'three body' problem? Isn't this like the famously difficult problem of attempting to model the simultaneous collision of three snooker balls? If you consider taking the collisions two objects at a time, separated by an instant, you get a different answer depending on the order in which you examine the 'sub collisions'.

 

[arildno]

Won't you run into the classic, 'three body' problem? Isn't this like the famously difficult problem of attempting to model the simultaneous collision of three snooker balls? If you consider taking the collisions two objects at a time, separated by an instant, you get a different answer depending on the order in which you examine the 'sub collisions'.

Mmm..possible enough.
Which shows that the scope of the utility of such a modelling is non-trivial to determine (and might be said to be rather more limited than we would like);
however we might, at first (for the problem I proposed) regard the impulses acting on the cube from the rod and the floor to act strictly simultaneously.

This means that we assume that the edge of the cube in contact with the floor on the opposite side of where the rod strikes can never have a non-zero collision velocity relative to the floor, i.e, the vertical component of the edge velocity must be required to be zero throughout the collision (given that the floor has no velocity).
The only point on that edge where we can regard the vertical impulse to act on (i.e, the average location), must be the midpoint on that edge, since otherwise, a torque would be generated about the \vec{i} axis, propelling some portion of that edge into the floor.

 

[NoTime]

i.e, the vertical component of the edge velocity must be required to be zero throughout the collision (given that the floor has no velocity).

That could be true if the collision lasted less time than the time it take sound to propagate thru the cube.
However, I don't think the collision would be elastic during this period.