Impulse on Rubber and Metal Hammers

[jack action]

Thought experiment: push nail a short way into ground; drop a pingpong ball onto the nail, and repeat that 1,000,000 times. How much energy have you expended in pingpong KE? How much further has the nail gone in?

The OP's question is about the effectiveness of a material for the hammer versus another one. That is all about energy. Less effective means that some of the input energy was not used into driving the nail.

 

[haruspex]

Less effective means that some of the input energy was not used into driving the nail.

That doesn’t make it "all about energy". That would be true if all you needed to know to determine the distance the nail advances is the energy delivered by the hammer.
You can deliver the same energy with a rubber hammer as with a steel one. The question is why the steel hammer is more effective for the same energy.
You could equally claim it is all about momentum - after allowing for the momentum that gets wasted.

The key to understanding why you need the hammer to be made of a stiff material is thinking about static friction and the force threshold that has to be exceeded.

 

[jack action]

You can deliver the same energy with a rubber hammer as with a steel one.

You can deliver the same [ouput] energy with a rubber hammer, but it requires more [input] energy. Hence why it is less effective. With a rubber hammer, my body does more work (energy) for the same result. The extra energy is wasted into elastically deform the rubber hammer.

You could equally claim it is all about momentum - after allowing for the momentum that gets wasted.

With another type of material, the hammer could break or be permanently deformed and it wouldn't bounce back and you could even have no - or very little - wasted momentum. It would still be less effective because it has wasted energy to break or deform the material.

Why does the hammer bounces back and gets that extra momentum? Only the energy stored in the hammer while it compresses explains where this momentum comes from.

The key to understanding why you need the hammer to be made of a stiff material is thinking about static friction and the force threshold that has to be exceeded.

I already explained that in post #34 and cited you as a reference, so I understand what you are saying. But this threshold force doesn't depend on the material of the hammer. It doesn't matter if the hammer is stiff or not. If it can deliver the force, the nail will move. How far will it move (work done, ##Fd##) depends on the energy transferred from the hammer to the ground. To evaluate this energy transfer, the stiffness of the hammer has an influence, as demonstrated in post #34. Heck, the stiffness of the nail is just as important too.

 

[jbriggs444]

It doesn't matter if the hammer is stiff or not. If it can deliver the force, the nail will move.

If the hammer is stiff, it can deliver more force for a given impulse. If it is not, it cannot deliver as much.

 

[jack action]

It doesn't matter if the hammer is stiff or not. If it can deliver the force, the nail will move.

If the hammer is stiff, it can deliver more force for a given impulse. If it is not, it cannot deliver as much.

The only reason why a rubber hammer wouldn't deliver the same force as a steel hammer is if the rubber hammer would store all its kinetic energy while deforming (into elastic energy) before it would be able to reach the threshold force required to break the ground. But if it can reach the threshold force, it's a maximum, nothing more is needed of the hammer, except energy. This is what I meant.

 

[jbriggs444]

The only reason why a rubber hammer wouldn't deliver the same force as a steel hammer is if the rubber hammer would store all its kinetic energy while deforming (into elastic energy) before it would be able to reach the threshold force required to break the ground. But if it can reach the threshold force, it's a maximum, nothing more is needed of the hammer, except energy. This is what I meant.

I am still not certain what point you are trying to make here. Even if a rubber hammer can succeed in delivering enough force during some portion of an elastic collision to cause forward acceleration by the nail this does not provide an assurance that the total amount of kinetic energy delivered to the nail during that elastic collision exceeds that from a rigid but inelastic collision.

In particular, the phrase "it's a maximum" conveys nothing to me. Exactly what thing is a maximum out of what set of things?

We can agree that if the force delivered exceeds that of static friction at some point, this does guarantee that the collision has succeeded in contributing some energy to the nail and that the nail has accordingly made some progress into the work.

 

[TSny]

Here's a video. Not sure it's all that relevant to the question at hand. But, it's fun to watch

 

[gmax137]

If you hit a nail with a rubber mallet, all that will happen is, the rubber head will get stuck on the nail. You will drive the hammer head "around" the nail, since the rubber is soft and the nail isn't.

I have quite a few hammers, ranging from lead, rubber, plastic, rawhide, copper, brass, steel... from an ounce to 8 pounds. They all have their uses. Driving nails is not one of the rubber mallet's strong points.

I just did a little fooling around with various hammers. I am beating them on the flat surface of a vise.

My rubber mallet has a big head and the rubber is pretty hard. It does not "bounce" much, if at all. It is more of a "dead blow" tool. Another (rubber) hammer has a smaller diameter head and the rubber is softer, this one bounces a little, but not much. The steel ball-pein bounces more, a lot more.

I think the entire premise of this thread ("rubber bouncy") is off-base.

 

[haruspex]

if it can reach the threshold force

Well, that's a big if for a start. So you do accept that a stiffer hammer is better able to exceed the threshold, and if the threshold is not exceeded the nail does not advance?

Next, what happens if the threshold is exceeded, but only when the rubber is almost at maximum compression, so has nearly come to a stop? Yes, there's energy stored in the rubber, but what happens to it?
@gmax137 makes the point that we've all been assuming rubber is more elastic than steel, in the sense of returning a higher fraction of stored energy, but that may well be false. But even supposing it is true, what will happen to that energy?
The nail starts to advance, but the hammer now reaches instantaneous rest and starts to rebound. Both the small advance of the nail and the hammer's rebound reduce the compression force, making it fall back below the threshold. The remaining recoverable energy in the hammer goes into the rebound. Very little has gone into advancing the nail.

 

[haruspex]

If you hit a nail with a rubber mallet, all that will happen is, the rubber head will get stuck on the nail. You will drive the hammer head "around" the nail, since the rubber is soft and the nail isn't.

I have quite a few hammers, ranging from lead, rubber, plastic, rawhide, copper, brass, steel... from an ounce to 8 pounds. They all have their uses. Driving nails is not one of the rubber mallet's strong points.

I just did a little fooling around with various hammers. I am beating them on the flat surface of a vise.

My rubber mallet has a big head and the rubber is pretty hard. It does not "bounce" much, if at all. It is more of a "dead blow" tool. Another (rubber) hammer has a smaller diameter head and the rubber is softer, this one bounces a little, but not much. The steel ball-pein bounces more, a lot more.

I think the entire premise of this thread ("rubber bouncy") is off-base.

We can avoid the nail penetrating the rubber hammer by supposing a broad flat head.
But yes, we should not assume a rubber mallet bounces more than a steel one. It will depend on the nature of the rubber.

That said, I suspect whoever set the question was under the same delusion regarding relative bounce, so to appreciate the intent of the question we should compare a stiff non-bouncing hammer with a soft bouncing one.

 

[jack action]

In particular, the phrase "it's a maximum" conveys nothing to me. Exactly what thing is a maximum out of what set of things?

I made a mistake here. As I said in post #34, I was imagining that if the resistive force requires, say, a 100 N before the nail moves, then this would be the maximum force that will exist in the system. Even if you hit the nail "harder", the force will still be 100 N, because the resistive force does not increase when you hit "harder". I was wrong because the nail will go in faster; which means an acceleration of the nail, which means the input force must be greater than the friction force.

Still it doesn't change my analysis based on energy, where the kinetic energy of the nail is equal to the kinetic energy of the hammer less the energy required to deform the different components (the ground, the nail, the hammer). The more deformation there is, the less energy is available to move the nail.

----------------------------------------------------​

Well, that's a big if for a start. So you do accept that a stiffer hammer is better able to exceed the threshold, and if the threshold is not exceeded the nail does not advance?

Yes.

Next, what happens if the threshold is exceeded, but only when the rubber is almost at maximum compression, so has nearly come to a stop? Yes, there's energy stored in the rubber, but what happens to it?
@gmax137 makes the point that we've all been assuming rubber is more elastic than steel, in the sense of returning a higher fraction of stored energy, but that may well be false. But even supposing it is true, what will happen to that energy?
The nail starts to advance, but the hammer now reaches instantaneous rest and starts to rebound. Both the small advance of the nail and the hammer's rebound reduce the compression force, making it fall back below the threshold. The remaining recoverable energy in the hammer goes into the rebound. Very little has gone into advancing the nail.

But you are repeating exactly what I said in post #34:

And when all the energy has been disposed of, the nail stops, and the elastic energy stored into the hammer comes back out. But at that point, the impact force will go from 100 N to 0 N as the spring (i.e. hammer) goes back to its original shape. So the nail will not get driven into the ground, and the stored energy will be converted to pure kinetic energy for the hammer. The amount of energy stored in the spring will determine how high the hammer will bounce back.

Did anyone read it?

 

[haruspex]

But you are repeating exactly what I said in post #34:

So why do you keep saying it is "all about energy"??
And why do you write

. It doesn't matter if the hammer is stiff or not.

?

 

[jbriggs444]

But yes, we should not assume a rubber mallet bounces more than a steel one. It will depend on the nature of the rubber.

The question of elastic versus inelastic is much ado about nothing.

If the nail is much less massive than the hammer head then the collision between the two does not waste much energy regardless. The collision energy of nail with hammer scales with the mass of the nail. It is a small fraction of the kinetic energy in the hammer head.

If the distance by which the nail advances is large compared with the deformation of nail and hammer upon impact then the loss of kinetic energy to that deformation also becomes irrelevant.

But -- if that deformation of the hammer head becomes significant relative to the advance of the nail, then that deformation, elastic or not, results in lost energy.

Stiff beats soft regardless.

 

[jack action]

So why do you keep saying it is "all about energy"??

Because I'm only talking about energy in my explanation.

Although, I've been waiting on your explanation for a few days now; just to see how you can explain it any other way.

I don't understand what you expect to your question in post #3 and what you will do with that answer:

Which are you as user supplying, force or momentum?

(I would say kinetic energy)

Also, I'm not sure what you are trying to prove with post #20:

You are not considering what opposes the advance of the nail. We are driving it into a fixed block. To make progress we have to overcome the resistive force.

After 24 posts over a period of 2 days, that is in when I chimed into bring something that would push the discussion forward a little bit.

So how do you explain that the rubber hammer is less effective than the steel one?

 

[haruspex]

Because I'm only talking about energy in my explanation.

That you are only talking about energy is not an explanation for why you are only considering energy.

I don't understand what you expect to your question in post #3 and what you will do with that answer:

Since we are assuming the hammers are equal mass, it doesn't matter whether you are supplying energy or momentum. I could have written either. My point was that you are not supplying force directly. But it is force that is needed to overcome static friction, so what matters in the characteristics of the hammer is how well it translates its energy/momentum into a large force, i.e. its stiffness.

So how do you explain that the rubber hammer is less effective than the steel one?

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

If, as you claim, it does not matter whether the hammer is stiff, how do you explain that of two hammers with the same mass, energy and momentum, steel does better?

 

[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.