"g" paradox...

[iacob alex]

  ...you can watch at :   http://www.youtube.com/watch?v=NFJPQLhWPvA    with the title "Falling Meter  Stick With Coins".

    In a classical lab-demo (Hinged stick-falling ball),you can see the acceleration paradox,when a ball and a board are released together,the end of the board outstrips its center of mass,leaving the board behind.

    That is,if you allow a board to rotate under the influence of gravity,the free end will accelerate at a rate greater than "g".

    We know that,gravity accelerates all objects at same rate...on the vertical line.

    This time,the relation between angular acceleration and linear acceleration,gives us a free-fall paradox (you remind the theory about the closed loop in gravity and the null result...).

    Here can be an "open gate" for our topic :if between two points,on the vertical line,for the same mass,we can obtain two different falls (velocity,acceleration,momentum....)...their inequality,can be a starting point for us...a "heavy angular fall"  vs.  an "easy linear up".

      All the Bests!  /  Alex   

[Yucca]

Hi Alex,

I think it happens because the ruler is rigid and fixed at one end. The middle of the ruler wants to fall at freefall speed. But because one end is restrained it cant fall at freefall speed. The difference is transmitted as torque to the free moving end through rigid lever action and so the end falls faster than freefall speed. Conversely the end that is now falling faster than freefall will be exerting a negative torque, trying to slow the whole thing down.  Only one point on the ruler falls at freefall speed, every other point being either faster if further away from the fulcrum or slower if closer. The whole system finds force equilibrum, by redistributing the kinetic energy.

In my mind it´s not a paradox, sadly I think if one analysed the movement and applied integral calculus then it would be net neutral so the whole system kinetic energy would be the same as if the ruler were dropped half the height with both ends free, of course I could be wrong. The simplest way to test would be with impact analysis using a really rigid beam with a bit of weight to it:

Test1:
allow a 1m beam to swing downward, one end travelling 40cm. The other end travelling 0cm. Analyse the impact force by seeing how far a pencil can be pushed into a slab of plastiscine by the fast moving beam end.

Test2
allow a 1m beam to drop freely, neither end restrained, through 20cm. Analyse the impact force by seeing how far a pencil can be pushed into a slab of plastiscine by the middle of the beam.

I think in both tests the resulting impact will be the same when many drops are performed to eliminate noise.

[iacob alex]

       Hi  Yucca !

  The term "paradox" is not mine...you can encounter on net.

  Paradox is a statement that seems absurd or self-contradictory,but may be true.

  You can watch short movies on net with the tag "Falling chimney",regarding the same puzzle.

  If you type "Falling stick" on net,you can find a plenty of information:it's simply classic lab-demo in elementary physics.We need proof no more ...the phenomenon is real...nothing new under the Sun!

         All the Bests! / Alex

[Yucca]

Hi Alex,

It doesn't seem absurd or contradictory to me, just normal rigid body dynamics in action. But maybe it's a paradox to some?

best, Yucca.

[Yucca]

My explanation above is somehwat abstract, it's difficult to explain it simply. But imagine this thought experiment, it will make things alot clearer:

Take a 1m long very rigid and lightweight stick, say a thin carbon fibre pole.

Pivot one end of the stick on the edge of a table.

Now 25cm from the pivot point attach a length of string, tied to the stick.

On the free end of the string hang a 1kg weight.

Now lift the sticks free end until it is horizontal, the 1kg weight hanging on the string below.

Now release the stick....

The 1kg weight will fall at nearly 1g, but of course we all know the end of the pole, 1m away from the pivot, will be accelerating vertically alot faster than 1g because it has to cover more distance. The end is being forced down by the inboard weight, acting through the rigid body of the stick, forcing the sticks tip to accelerate well beyond 1g.

This is exactly the same as a uniform ruler doing the same thing, only in my thought experiment above the effect is exaggerated for easier visualisation.

The whole system acts perfectly conservatively, no magic here unfortunately.