Rigid Body dynamics - elastic collisions

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Homework Help Overview

The problem involves a uniform rectangular tile dropping and colliding elastically with a vertical pole. The scenario requires analyzing the tile's velocity, angular momentum, and angular velocity immediately after impact, while considering the conservation of kinetic energy and angular momentum.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the conservation of kinetic energy and angular momentum, questioning how to apply these principles to find the necessary quantities. There are inquiries about the moment of inertia tensor and its role in the calculations, as well as concerns about determining the axis of rotation and angular velocity.

Discussion Status

Some participants are sharing equations and discussing the moment of inertia tensor, while others express uncertainty about their approach. There is an ongoing exchange of ideas, with guidance being offered on writing out equations and applying the parallel axis theorem.

Contextual Notes

Participants are working under the constraints of the problem statement, which includes the assumption of an elastic collision and the requirement to find specific quantities immediately after impact. There is also a mention of external torque-impulse considerations.

Plutoniummatt
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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...
 
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Hi Plutoniummatt! :smile:

(what is the velocity just before impact? on my computer it looks like (0; 0;u) :confused:)
Plutoniummatt said:
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.
 
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
 
Hi Plutoniummatt! :wink:

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

(and yes, you will need the moment of inertia tensor for the angular momentum equation)
 
Hi Plutoniummatt! :smile:

(have an omega: ω :wink:)

Let's see … your moment of inertia tensor Îc (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. :smile:
 

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