Impulse on Rubber and Metal Hammers

[jack action]

At the risk of repeating myself, steel wins by virtue of its stiffness.

This is not an explanation, it is just an observation. Why do stiffness matters? How can you quantify the effectiveness of a hammer versus another one?

If, as you claim, it does not matter whether the hammer is stiff,

Never said that, as you mean it in that sentence. I said that no matter its stiffness, a hammer can produce any force desired. Therefore a nail can go into the ground, no matter what type of hammer you are using. The proof of that is that if you put a hammer on the nail and push on the hammer with a piston, the hammer will transfer the force onto the nail no matter how high it is. The stiffness is irrelevant to how much force it can transmit.

how do you explain that of two hammers with the same mass, energy and momentum, steel does better?

Answering my own question: How do I quantify the effectiveness of a hammer versus another one?

What do I want to achieve? I want to move a nail of a given length (say 0.1 m) into the ground which offers a certain resistance (say 100 N). I know that I need 10 J of work to be done. If I was using a tool that provided 100 % effectiveness, I would expect putting 10 J of energy into that tool. If I put 10 J of kinetic energy into a hammer and the nail goes down 0.05 m, I would say the hammer is 50 % effective.

Where did the lost energy go? That is the question I answered in post #34. If it goes into elastic energy, it comes back into the form of kinetic energy in the hammer. If we have plastic deformation, then it is transformed into heat.

I don't see how you can estimate (or appreciate) the effectiveness of a hammer vs another one with momentum or force. How much momentum does it take to drive a nail 0.1 m, starting at rest, ending at rest? How much force does it take to do the same? That is easy, 100 N. But we're not considering the distance traveled, which is important. And as I said earlier, any material can provide any force (assuming only deformation, no breakage).

You even mention 'mass' in your question. Does a hammer's mass matter? No it doesn't. If you have a smaller hammer, you can still put 10 J of kinetic energy into it, and the nail will go down just the same. Of course, if you are limited on the velocity you can transfer to the hammer, then you may not be able to get to 10 J and a bigger hammer will be necessary. But without such limit, mass doesn't matter to evaluate its effectiveness.

 

[jbriggs444]

Does a hammer's mass matter? No it doesn't

Yes, it does. They make different size hammers for a reason.

 

[jack action]

Yes, it does. They make different size hammers for a reason.

The reason being that a human cannot physically - and effectively - set the hammer to any velocity. There is a limit.

 

[jbriggs444]

The reason being that a human cannot physically - and effectively - set the hammer to any velocity. There is a limit.

Right. So we agree that the hammer's mass matters.

 

[haruspex]

I said that no matter its stiffness, a hammer can produce any force desired.

The whole point of the question is to compare the effectiveness in pushing a nail into the ground of two hammers of the same mass and swung at the same speed. We all, I hope, expect the steel hammer to be more effective than the rubber hammer; the question is why.
My answer is stiffness. What is yours?

 

[Delta2]

What do I want to achieve? I want to move a nail of a given length (say 0.1 m) into the ground which offers a certain resistance (say 100 N). I know that I need 10 J of work to be done. If I was using a tool that provided 100 % effectiveness, I would expect putting 10 J of energy into that tool. If I put 10 J of kinetic energy into a hammer and the nail goes down 0.05 m, I would say the hammer is 50 % effective.

Where did the lost energy go? That is the question I answered in post #34. If it goes into elastic energy, it comes back into the form of kinetic energy in the hammer. If we have plastic deformation, then it is transformed into heat.

I don't see how you can estimate (or appreciate) the effectiveness of a hammer vs another one with momentum or force. How much momentum does it take to drive a nail 0.1 m, starting at rest, ending at rest? How much force does it take to do the same? That is easy, 100 N. But we're not considering the distance traveled, which is important. And as I said earlier, any material can provide any force (assuming only deformation, no breakage).

Before we can talk about effectiveness in terms of energy, the nail must start moving (otherwise no work, no transfer of energy occurs), which means that the force by the hammer on the nail must be greater than the resistance force from the wall or the floor. So we first have to talk about effectiveness in terms of force. And that's where all the talk about momentum, impulse and time of contact comes. And no i don't think all materials have the same effectiveness in terms of force, rubber hammers having much greater time of contact, apply much less average (over time) force to the nail, so the nail probably won't move at all if it is put against a hard surface.

 

[jack action]

rubber hammers having much greater time of contact, apply much less average (over time) force to the nail

But why? The answer is: because there is an energy transfer required to deform the hammer.

There are two ways to look 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.

The first one sounds way more logical: everything begins with an energy loss. The smaller force is a consequence of this, not a cause.

As I said before, compress a spring (the hammer) enough, and you will obtain any force you want. But why can't you achieve the same force with a rubber hammer as a steel hammer? Why can't you just compress it a little bit more? Because of the energy loss.

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

For example, the theory behind inelastic collision and the coefficient of restitution, uses conservation of momentum and conservation of energy. Nothing about forces. The coefficient of restitution is defined as a ratio of energy. This theory doesn't tell us how or why the energy wasn't totally transferred to the nail (there's nothing about material stiffness either), but it tells you the end result anyway. What is it looking at? Oh yeah: Energy, energy, energy, energy, energy, ...

Your best bet without referring to energy is looking at the impulse. If the only thing you have is the duration of the impulse, you can get an average force over that period of time. How much time did it spend over the threshold force (if it did)? Nobody knows. You have to know the force vs time function. You have to model the hammer as a spring and plot that function considering the initial velocity. But how do you evaluate the effectiveness with such a theory? The ideal duration would be zero? Yes, you know that if one hammer takes 2 seconds and the other takes 1 second, the latter is more effective. But is 1 second good or bad to begin with? We don't know since the ideal time is zero. Another question: Where did the 'lost' force go? A question that I'm not even sure makes sense.

The conservation of energy principle is so much more elegant, complete and easier to visualize.

 

[Delta2]

But why? The answer is: because there is an energy transfer required to deform the hammer.

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.

The first one sounds way more logical: everything begins with an energy loss. The smaller force is a consequence of this, not a cause.

Sorry i don't understand why the energy loss required to deform the rubber hammer results in smaller force. Can you expand on this? My intuition tells me that the energy loss due to the deformation of the rubber hammer ,which seems negligible, has little to do with the much smaller force on the nail.

For me the sequence of events goes like this :
deformation of the rubber hammer->greater contact time->smaller average force via the impulse/contact time quotient.

 

[jbriggs444]

Sorry i don't understand why the energy loss required to deform the rubber hammer results in smaller force. Can you expand on this? My intuition tells me that the energy loss due to the deformation of the rubber hammer ,which seems negligible, has little to do with the much smaller force on the nail.

For me the sequence of events goes like this :
deformation of the rubber hammer->greater contact time->smaller average force via the impulse/contact time quotient.

[Note that I do not want to be drawn into a bicker war about energy versus momentum]

It takes less force to dissipate the same energy in a softer object. Let us idealize the situation a bit and use Hooke's law to see why.

We have a fixed energy E to dissipate. We have a hammer whose stiffness is k. The constant k is expressed in units of force per unit deformation. We want to calculate the maximum force involved.

We start by calculating the required deformation to dissipate the energy. Call the deformation s.$$E=\frac{1}{2}ks^2$$Solving for s, that gives$$s=\sqrt{\frac{2E}{k}}$$The maximum force is given by$$F=ks=k \sqrt{\frac{2E}{k}}=\sqrt{2Ek}$$Which is to say that maximum force scales as the square root of stiffness.

Or, more pithy: It takes less force to dissipate the same energy if you deflect farther.

 

[Delta2]

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?
The way i knew this was via impulse and contact time, but now i see it can be explained via energy and stiffness. Which explanation is primary or better in your opinion?

 

[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.