A weird collision theorem

In summary, the professor was discussing 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. He found that if he lifted a 25 lb weight plate and released it suddenly, the scale would momentarily read near 50 lbs. He is wondering if this is a result of the beam being a spring-loaded scale, with some finite dx.
  • #1
HoloBarre
7
0
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
 
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  • #2


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


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


HoloBarre said:
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
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


HoloBarre said:
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


Twice the force, half the time?
 

1. What is the "weird collision theorem"?

The "weird collision theorem" is a scientific principle that describes the outcome of a collision between two objects that have different masses and velocities. It states that in an elastic collision, the total momentum and kinetic energy of the two objects will be conserved, but the individual velocities of the objects after the collision may not be equal to their initial velocities.

2. How is the "weird collision theorem" different from other collision theorems?

The "weird collision theorem" differs from other collision theorems in that it takes into account the masses and velocities of the colliding objects, rather than assuming they are equal. This allows for a more accurate prediction of the outcome of the collision.

3. What are some examples of "weird collisions"?

Some examples of "weird collisions" include a small object colliding with a much larger object, a fast-moving object colliding with a slow-moving object, or two objects with vastly different masses colliding at an angle.

4. How is the "weird collision theorem" applied in real-world situations?

In real-world situations, the "weird collision theorem" can be used to understand and predict the outcome of collisions between objects of different masses and velocities. This can be useful in fields such as engineering, physics, and sports.

5. Can the "weird collision theorem" be applied to non-elastic collisions?

Yes, the "weird collision theorem" can also be applied to non-elastic collisions, where some kinetic energy is lost during the collision. In these cases, the total momentum of the system is still conserved, but the individual velocities of the objects after the collision may be even more different from their initial velocities.

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