Rigid Body dynamics - elastic collisions

[Plutoniummatt]

Homework Statement

A uniform rectangular tile drops without spinning until its corners reach positions
(0; 0; 0); (2a; 0; 0); (2a; 2b; 0); (0; 2b; 0), when it strikes the top of a vertical pole at a
point very close to the (0; 0; 0) corner. Just before impact the velocity of the tile was
(0; 0;u). Assuming that the tile does not break, and that the impact is elastic (i.e.
the kinetic energy of the tile is conserved), find immediately after impact

(a) the velocity of its centre;
(b) the angular momentum about its centre;
(c) its angular velocity.

Show that the velocity of the corner at (0; 0; 0) becomes (0; 0;+u) immediately after
impact.

Homework Equations

Inertia tensors of the rigid body involved, all standard mechanics equations.

The Attempt at a Solution

I honestly have no idea how to approach this...all i can write down is the initial KE = final KE...

 

[tiny-tim]

Hi Plutoniummatt!

(what is the velocity just before impact? on my computer it looks like (0; 0;u) )

I honestly have no idea how to approach this...all i can write down is the initial KE = final KE...

ok, write that down.

Then, since there's no external torque-impulse about the corner, write down conservation of angular momentum about the corner.

 

[Plutoniummatt]

Ok, what do I do after? Something to do with moment of inertia tensor maybe? although don't know how to find the axis of rotation...and how do I find angular velocity?

Thanks

 

[tiny-tim]

Hi Plutoniummatt!

Write out all the equations, and see where you get.

(and yes, you will need the moment of inertia tensor for the angular momentum equation)

 

[Plutoniummatt]

Hi Tim,


Am I going in the right direction?


http://img641.imageshack.us/i/69003327.jpg/
http://img535.imageshack.us/i/87631937.jpg/


http://img535.imageshack.us/i/87631937.jpg/
http://img641.imageshack.us/i/69003327.jpg/

 

[tiny-tim]

Hi Plutoniummatt!

(have an omega: ω )

Let's see … your moment of inertia tensor Î_c (about the centre of mass) has principal moments of inertia mb²/3 ma²/3 and m(a² + b²⁾/3.

So yes, now use the parallel axis theorem to get the principal moments of inertia at (0,0,0), and then use Îω and ωÎω/2.