# A weird collision theorem

1. Dec 23, 2008

### HoloBarre

A weird collision "theorem"

All --

Many moons ago I thought I heard a professor refer to a "theorem" in mechanics which stated that if a hard object is released from an infinitesimal height above a hard surface, the force will be exactly twice the weight of the object.

Did I hear correctly? If so, any ideas on a derivation of this?

Thanks

2. Dec 23, 2008

### Loren Booda

Re: A weird collision "theorem"

Observation: any hard object resting "on" a hard surface would seem to be "released" from an infinitesimal height.

A strange question: do the surface and object hardnesses approach infinity faster than the height approaches zero?

Paradox: for the stated collision to conserve energy and momentum, the surface must approach infinite mass and thus impart a nearly infinite gravitational force to the object, thus implying a nearly infinite object weight.

The above is somewhat speculative, and should not be considered too seriously - but please, share where I err.

3. Dec 24, 2008

### HoloBarre

Re: A weird collision "theorem"

I think part of the problem is that energy/momentum/collision equations say little or nothing about forces themselves, but only as a force in integrals of Fdt and Fdx. Thus, F by itself is poorly constrained.

However, by making dx, dt, and V all essentially zero, maybe now somehow F is tightlyh constrained or determined.

Toward this end, I took a 25 lb. weight plate, and a large-dial bathroom scale, and weighed the plate. Unsurprisingly, it weighed just about 25 lbs (expensive scale!).

I then "lifted" the plate *just to the point* where the scale again read zero, and then released the plate as abruptly as possible. And bang, the scale momentarily read near 50 lbs.... or so I think.

Dropping the plate from just a fraction of an inch higher would send the needle wildly up, but just at the surface the process seemed to give a near-50lb reading.

Could it be me "forcing" a desired result? Possibly.
Could this be the artifact of a spring-loaded scale, with some finite dx?? Possibly, but it seems to me that *something* is in fact going on here, ito the above "theorem".

4. Dec 24, 2008

### Staff: Mentor

Re: A weird collision "theorem"

Hi HoloBarre, welcome to PF,

What you are talking about is called a "suddenly applied load", and it is not the force that is twice the weight, but rather the strain is twice the statically loaded strain. I don't remember the derivation exactly but I do remember that it requires a few assumptions: there are no energy losses due to plastic deformation or heat etc. (energy in gravity, elastic, or kinetic), the beam is not very massive (no change in PE due to beam movement), the strain is uniform throughout the beam, and maybe one or two others I can't remember.

Basically, consider the beam to be a spring (Hookes law). The statically loaded strain is an equilibrium position where the stress exactly balances the weight. When the weight is suddenly applied you see that it is not in equilibrium, the stress is less than the weight, so the mass accelerates and the beam strains. As the beam reaches the equilibrium strain the forces balance, but now the mass has kinetic energy. The mass continues moving and the beam continues straining and now that it is beyond the equilibrium strain the stress is greater than the weight so the mass begins to decelerate. Eventually, the kinetic energy is entirely converted to elastic energy and the beam has reached its maximum strain. This strain is twice the equilibrium strain. The stress is still greater than the weight so the mass will continue to accelerate back up and will vibrate until dissapative forces eventually cause it to come to rest at the equilibrium strain.

5. Dec 26, 2008

### Phrak

Re: A weird collision "theorem"

Twice the force, half the time?