Impulse on Rubber and Metal Hammers

[jbriggs444]

ok @jbriggs444 thanks for the analysis. But i don't understand what is the energy E? Is it the kinetic energy of the hammer before the collision? Or just the portion of kinetic energy that goes as energy required to deform the hammer?

As is, it is an analysis of an idealized impact of a perfectly elastic hammer with an unyielding surface. One could re-do the analysis to determine the energy absorbed from a hammer before the nail starts to budge. It'll likely be messier.

So we have a hammer with energy E and stiffness k. And we have a target that resists our impact, remaining immobile until subject to force F.

We begin by determining how far the hammer deforms when the target it just ready to budge.$$s=F/k$$The energy absorbed into deformation of the hammer is given by $$E_d=\frac{1}{2}ks^2 = \frac{1}{2}k(F/k)^2 = \frac{1}{2}F/k$$The massless nail and deformed hammer now advance together into the work against resisting force F until they decelerate to a stop. The deformation energy in the hammer is then released as the hammer rebounds.

One can see that the energy lost to deformation in the hammer is inversely proportional to k in this case. Stiffer is better. However, it is also directly proportional to F. For a very loosely fitting nail, the loss may be negligible. For a tight fitting nail it may exceed 100% and the nail may remain stubbornly in place.

Which explanation is primary or better in your opinion?

That is a disagreement that I want to tip-toe quietly past.

 

[haruspex]

There are two ways to like at it:

1. The deformation of the hammer takes some energy away, which translates into a smaller force pushing the nail;
2. The hammer deformation takes some force away, which translates into an energy loss.

If we take the rubber hammer as perfectly elastic there is no energy loss. Rather, the delivery of the same energy gets spread over time. You cannot explain why that makes it less effective without considering forces.
Or compare striking the nail once with a steel hammer with a certain KE, and striking it ten times with a tenth the KE.

You can even explain the phenomena without looking at the force provided.

Suppose there is no static friction, e.g. we are striking a nail laid horizontally. The hammer with the greater elasticity wins. Within bounds, the stiffness ceases to matter because there is no threshold resistance to overcome.

 

[jack action]

If we take the rubber hammer as perfectly elastic there is no energy loss. Rather, the delivery of the same energy gets spread over time. You cannot explain why that makes it less effective without considering forces.
Or compare striking the nail once with a steel hammer with a certain KE, and striking it ten times with a tenth the KE.

Suppose there is no static friction, e.g. we are striking a nail laid horizontally. The hammer with the greater elasticity wins. Within bounds, the stiffness ceases to matter because there is no threshold resistance to overcome.

Thank you for the nice response useful to the discussion. I agree with that. This is why I liked my analysis in post #34 where I was identifying where the energy goes; and yes, I need to consider the threshold force to determined how much energy is lost.

There is also a case I thought of where stating the effectiveness depends on the material stiffness can be misleading. Two hammers made of steel, same mass, same material stiffness, same input velocity, but one is more effective than the other, why? Because they don't have the same design. One is a full cylinder of steel, the other is a thin tube of steel between two steel plates, where the tube part can be crushed under the threshold force. All of this because it takes energy to deform the hammer that cannot be used to push the nail.

In the automotive industry, it took something like 50 years to understand that a soft bumper was better than a stiff one. Even though this theory of collision was well defined at the time. But looking at only forces, people just imagine that a stiffer bumper would give more force and therefore better protect the passengers. That is until someone realized that a soft bumper - especially if it deforms permanently - absorbs energy that the passengers don't have to absorb themselves. The moral of the story is that it is important to understand where the energy goes.