Why Does Distance from a Pivot Point Reduce Effort in Mechanics?

[tiny-tim]

didn't robert frost write a poem on how going a shorter distance gives a greater force? …

i took the path less travelled

and that made all the difference​

 

[sophiecentaur]

There are some things which just do not have explanations within a limited set of knowledge. One can't just 'demand' the terms that can be used.
No one would try that sort of thing with Maths or Software so why assume it can work with Physics?

 

[kjamha]

First off, I want to apologize for not being as technical as I should be, but I'll do my best. Let's say your kid brother is on a see saw and he weighs 100 lbs. In order to raise him up 4', you need to push down your end of the see saw with a force of 100 lbs for 4 feet. Think of pushing down on your end of the see saw for 4 feet as a certain amount of energy that you need to exert. No matter what, if you want to raise your kid brother up 4', you need to exert that amount of energy. If you double the length of your end of the see saw, you can stretch out the amount of energy that you need to expend, it will be easier because you will push down with a smaller force - if fact, you will only have to push down with half the force, but it will take twice as long (you now need to push down for 8'). You still deal with the same amount of energy, but you don't have to exert all of that energy so quickly. You can spread it out - thin it out.

 

[zoobyshoe]

OK, mechadv24, I think I have figured out a good explanation of why distance from the fulcrum makes a difference without the use of any formulas and also avoiding any mystifying reference to "force multiplication". The explanation is a comparison of three different simple situations, and I even drew pictures:

Fig. 1

Figure 1 illustrates a stone block of weight 50 lbs supported on a massless board resting on two supports The board is understood to be level with the horizon, and the stone block is located with its center of gravity exactly over the center of the board. The board is exactly centered on the two supports. The stone block is of uniform size and density such that its center of gravity is congruent with its center of volume.

All these things being true, we should be able to conclude that each support bears exactly one half the 50 lb weight of the stone block, that is: 25 lbs per support.

Since each support bears half the weight, any thing substituted for one of the supports will also bear half the weight. If we substitute a man for one of the supports, that man will be maintaining a 50 lb weight off the ground by supporting 25 lbs of it himself and letting the other support bear the other 25 lbs.

The man might, then, lift or lower the 50 lb weight by what ever distance the other support allows, by only manipulating 1/2 of it himself. This is a second class lever (click and check out the little animated second class lever) :

http://www.elizrosshubbell.com/levertutorial/second.html

Fig. 2

In figure 2 the stone block has been pushed all the way over to the right. Its center of gravity is now directly over the center of the right hand support.

In this position the right support is bearing the full weight of the block. The left side support is holding none of the block. To prove this, we could remove the board, and the block would rest on the right hand support and not fall.

If the left hand support is holding none of the weight, then any thing we substitute for it would also be holding none of the weight. Likewise on the right. Any thing we substitute for the right side support will be supporting all of the weight, 50 lbs.

A man substituted on the left would be holding none of the block, and a man substituted on the right would be holding the total weight of the block.

Fig. 3

In figure 3 we have moved the block to a place between the positions illustrated in figures 1 and 2. Since the right hand support bore 1/2 the weight in fig. 1, and bore the full weight in Fig. 2, it must now, logically, be bearing something between half the weight and the full weight, based exclusively on the fact the block is now resting between the the two former positions.

By the same logic, the left hand support must now be bearing something between half the weight and none of the weight.

And, any thing we substitute for either support will be bearing the same fraction of the weight borne by the support it replaces.

If we put a man over onto the right, he will be supporting more than half of the weight but less than the full weight. If we substitute a man for the left hand support, he will be supporting more than none of the weight, but less than half of the weight.

The exact proportions are exactly what you'd think, but it is only necessary to show that the placement of the stone alters what each support must bear. We can arrange it, as I've shown, that both bear the same amount, or that one bears the full amount while the other none, and we can arrange for everything in between. As we push the block closer and closer to the right hand support, that support bears more and more of its weight. (Because if it doesn't, then there must be some threshold, some point where it suddenly changes from bearing half the weight to all of the weight. If you can prove there is such a threshold, why it should exist, and where it is, I think we'd all be amazed.)

As I said, this is a second class lever. We can turn it into a first class lever: If we lengthen the board in Fig. 3 past the right hand support by an amount equal to the distance between the center of gravity of the block and the right support, we can put the weight out onto this new extension, and a man over on the left will now have to press down with 1/4 the weight to sustain the weight. And, we will have turned the second class lever into a first class lever (Fig 4):

As before, the sturdy fulcrum is really providing most of the support (3/4 of it in this case) and the weight or person on the left must now push down where before they pushed up.

In conclusion, having viewed the weight on supports as a second class lever, and then having turned the second class lever into a first class lever, I hope I have demonstrated why the distance from the fulcrum at which the force is applied makes a difference. The lever is analogous to a weight resting on two supports, and, the relative distances of the supports from the center of gravity of the weight determine what fraction of the total weight a given support of that weight must bear. This same relationship follows in some way, shape, or form into the lever proper, and into all examples of mechanical advantage in all its manifestations.

 

[sophiecentaur]

The above is all fine but it makes the basic assumption of the Principle of Moments from the start - it's just not explicit. WHY should there be an equal share in the first example?
'Stands to reason', or the principle of moments?
Also, it involved an awful lot of writing and diagrams.
The point of expressing things in the way the Physics does is that it gets it all into just a line or two of argument / explanation - probably with some of the ultimate shorthand -Maths. We have been heading that way since Galileo's time, with good reason.

 

[JHamm]

But the OP doesn't want to just know that it IS true, I'm sure (s)he already knows that, what they're after is an explanation on how it can be true, what makes it true; while a few lines of maths is the best way to show that something is true, it doesn't make a very convincing statement about how or why. In this case the only way to explain to the OP what they're after is through diagrams because that is really what they're asking for.

 

[tiny-tim]

i'm with zoobyshoe on this

the original question was …

ok, I know that being further away and using a fulcrum/pivot point from an object being moved takes less energy. i.e using a 4 foot crow bar to pry open something. But i can't grasp the concept of why being further away makes it so much easier. thanks

… why does being further away make it easier?

zoobyshoe points out that if you're far enough away (but not too far! ), then the force needed is exactly zero …

it follows logically that in between, the force needed will be less, but not zero

i think that's a very good answer! ​

 

[sophiecentaur]

But the OP doesn't want to just know that it IS true, I'm sure (s)he already knows that, what they're after is an explanation on how it can be true, what makes it true; while a few lines of maths is the best way to show that something is true, it doesn't make a very convincing statement about how or why. In this case the only way to explain to the OP what they're after is through diagrams because that is really what they're asking for.

I can appreciate this to an extent but this concept of 'understanding' is actually not much more than reconciling a new idea with our already established ones. There is an enormous temptation to assume that the familiar is 'obvious, per se, and that the next step along the road is different. It's not; it's just not familiar yet.
A lot of people post questions on this forum which ask for explanations and there are always a lot of contribution answers that involve highly personal and quirky models. These sorts of answers can come from all sorts of directions which may be nothing like the start point of the original questioner's understanding. The more involved the answer, the more chance of it being taken 'wrongly' - and that could be wrong, either in the comprehension of what the contributor meant or actually wrong, because the Science is flawed.

Why is it that Maths is used so often in 'conventional explanations'? It's because there is least risk of misinterpretation. There is a common standpoint for both the asker and the responder. As Feynman frequently said (and he is GOD on this forum) - there are no real answers to the "WHY" question. There are levels of explanation available which can satisfy different levels of scrutiny.
If someone tries to give an answer to a 'why' question, they are duty bound to give a caveat ("this is my interpretation") unless their answer is straight out of some reputable source.

So, the idea of moments can be discussed with maths OR with arm waving BUT neither discussion will yield a totally bomb proof "why" answer. The best it can do is to satisfy the questioner in some way. However, the Maths answer will allow the questioner to extend the knowledge further but the arm waving one cannot be relied on to do so; it may totally lack substance.

A caveat, here. Maths is only a model, of course and can yield nonsense results if it isn't interpreted appropriately. That even applies to some very simple situations.

 

[sankalpmittal]

Ok , zoobyshoe ,

Here are the answers which I have figured out (which involve reasoning which I hope OP has been looking for) :

ANSWER 1 :

We know that centre of gravity is the point where total weight of body is supposed to act.
Let one arm of lever be longer and other be shorter. Then its quite obvious that total weight downward equals sum total of weight of all the atoms downward. So if one side arm is greater than other , then at that side already there are more atoms and hence greater downward weight. So we apply less additional weight for the both side to be in equilibrium.

Also centre of gravity will be at midpoint of lever and hence will dominate at side which has greater lever arm.

http://postimage.org/image/i5170vxjr/

ANSWER 2 :

All I can ask you is to meditate the idea of force (as said by jambaugh) which is work done per unit displacement ! What if you have lighter body and heavier body in which you are doing equal work. Obviously you apply less force on lighter one. Hence in case of lever , work input equals work output. Greater arm covers more arc distance so you apply less force and vice versa in case of shorter arm. (Why ? same analogy... where you return to relative masses on which force due gravity is acting which is mg and yes , arm distance is directly proportional to arc distance covered by corresponding lever arm.) Edit : Please anyone reply whether I am correct or not ! I have been posting this post thrice in this thread ! xD

 

[zoobyshoe]

Ok , zoobyshoe ,

Here are the answers which I have figured out (which involve reasoning which I hope OP has been looking for) :

ANSWER 1 :

We know that centre of gravity is the point where total weight of body is supposed to act.
Let one arm of lever be longer and other be shorter. Then its quite obvious that total weight downward equals sum total of weight of all the atoms downward. So if one side arm is greater than other , then at that side already there are more atoms and hence greater downward weight. So we apply less additional weight for the both side to be in equilibrium.

Also centre of gravity will be at midpoint of lever and hence will dominate at side which has greater lever arm.

http://postimage.org/image/i5170vxjr/

ANSWER 2 :

All I can ask you is to meditate the idea of force (as said by jambaugh) which is work done per unit displacement ! What if you have lighter body and heavier body in which you are doing equal work. Obviously you apply less force on lighter one. Hence in case of lever , work input equals work output. Greater arm covers more arc distance so you apply less force and vice versa in case of shorter arm. (Why ? same analogy... where you return to relative masses on which force due gravity is acting which is mg and yes , arm distance is directly proportional to arc distance covered by corresponding lever arm.)


Edit : Please anyone reply whether I am correct or not ! I have been posting this post thrice in this thread ! xD

The only real objection I had to your previous posts was bringing kinetic energy into it. You know Occam's Razor? "Entities should not be unnecessarily multiplied." The fact the arms have kinetic energy was an unnecessary complication in sorting things out.

What you posted above is fine with me, but I already understand it. Mechadv24 has some sort of interesting mental blind spot that prevents him from grasping it with the usual amount of explanation.

It's off topic under the present circumstances, but it occurred to me there probably is a real situation where you might want to analyze a lever in terms of kinetic energy, and that would be if you were making a trebouchet:

http://en.wikipedia.org/wiki/Trebuchet

where you want the kinetic energy of the long arm and its sling to be maximized. I haven't thought that through yet, though.

 

[jambaugh]

Yet I cannot find answer to OP's question. He wants to analyze molecular interactions ? In other words , I think he is looking for theoretical reasoning rather than mathematical deductions.

When did the OP last post? Are you sure his question has not yet been answered to his satisfaction?

 

[zoobyshoe]

When did the OP last post? Are you sure his question has not yet been answered to his satisfaction?

This was his last communique:

thanks. I'll try to understand this. i tried crowbarring some nails before starting thread..I won't give up now knowing that it IS understandable that the greater distance but less force on my end of the lever will move an object with all the concentrated force but for a shorter distance.

In other words, he's provisionally accepting it on faith.

 

[sankalpmittal]

When did the OP last post? Are you sure his question has not yet been answered to his satisfaction?

No I am not. The OP's question has been answered several times in this thread. I gave my answer in post #44. zoobyshoe gave the answer in post #39. But the main problem is that OP is finding those answers unsatisfiable though. He wants something else but I don't know anyways.

My answer in post 44 is more theoretical and reasonable than mathematical which I hope OP has been looking for. I think my answer is easily understood here because its purely physics and yes , there are no absurd complications inserted like kinetic energy (*types this sentence being addressed to zoobyshoe particularly*). In that answer , I have used only correct and well permitted terms to avoid misunderstanding and wrong judgments.

 

[mechaadv43]

I'm pretty sure zoobyshoe's post made it click for me. i need to digest the concept a little more to be certain for myself, but that's all i was wondering. thanks for drawing that all out instead of just like MS paint.

 

[zoobyshoe]

I'm pretty sure zoobyshoe's post made it click for me. i need to digest the concept a little more to be certain for myself, but that's all i was wondering. thanks for drawing that all out instead of just like MS paint.

Well, that's good news. Let me know if you hit any more snags.

I draw all the time so that's my preferred way of tackling it. Paint would have been foreign to me.

 

[mechaadv43]

TY all, I'm not thinking clearly lately.
Simply, the weight sets on the fulcrum and i just have to move my proportion. the wight is also tranfered into the lever, as example, if i push a seasaw that's weighted on one side: if i push it from the end, there's more weight stored in the lever as opposed to if i push if from right next to the pivot.

got confused with experaments like nail and crowbar because they also use a big other factor of concentrating my force into a small area around the nail which i overlooked. Or for cutting a tree half way at the base then pulling it down from the top. once the tree's pulled past its 'spring' the force goes into the cut and the wood itself becomes the fulcrum until it breaks.

 

[zoobyshoe]

TY all, I'm not thinking clearly lately. Simply, the weight sits on the fulcrum and i just have to move my proportion (plus pivot friction).

Yes! You are basically dividing the weight into four parts (in my drawings) and you only have to deal with one part yourself. The fulcrum bears the weight of the other 3/4.

Lifting the full weight 1/4 the distance you push is the same as lifting 1/4 the weight the full distance you push. The advantage of the lever is that it let's you handle the weight a quarter (or whatever fraction) at a time instead of having to lift the whole thing the whole distance all at once. In the end, you have done exactly the same amount of work, so there's no net gain.

experaments like nail and crowbar also use a big other factor of concentrating my force into a small area around the nail, which is easy to understand.

Yes. You are dividing the work it takes to pull the nail between yourself and the wood near the nail where the hammer presses down. The nail is the "weight" and you are making the board bear most of it while you apply some fraction of it. And, the force on the hammer comes out the other side as a much larger force on the nail because the distance over which it's applied is much shorter. It's been "concentrated" into a smaller distance.

Or for cutting a tree half way at the base then pulling it down from the top. once the tree's pulled past its 'spring' the force goes into the cut and the wood itself becomes the fulcrum until it breaks. most of the force is just figting the 'spring' too.

Yes. And the falling part of the tree, itself, becomes both lever and applied weight (force). The length of the tree (from fulcrum to center of gravity) vs the length of the cut is described by the Law of the Lever.

Now, having gone this circuitous route, I sudden thought of the ultimate "intuitive" demonstration that distance from the fulcrum matters, and kick myself for not having thought of it first:

Stand up and put your feet about a foot apart. Slowly shift your weight to one side until all your weight is on one foot. You will be able to physically feel the pressure gradient in your feet as one foot bears more and more of the weight and the other less and less. It's directly tied to the distance of your center of gravity from either foot.

It's so obvious I couldn't see it, but everyday physical experiences like this are why most people automatically "get" why the distance from the fulcrum makes a difference as soon as they hear it said.

 

[mechaadv43]

uh, I'm so confused still. sorry to be a burden with my apparent stupidity (unless what I'm trying to 'get' isn't all that simple.i might take an iq test when i figure this out to hopefully find I'm not as dumb as i feel!) Formulas don't help me much, but I understand the concepts now. It's something i need to 'get' out on my own, but hoping it can be guided. this isn't a big joke, i don't get it still.
I'm drawing a blank on a simple experament:
A large tupperware box & a container of cat litter tied to the end of a 7 foot pole.
Putting the pole on the tuperware and sliding the pole to and fro.
-when the distances of my hand and the litter to the box are equal, I'm pressing as hard as the litter weighs. that's very easy to understand.
-when the litter's closer to the box than my hand, it's easier to hold.
-When my hand's close to the box and the litter is far, I have to press down like 10X the weight of the litter to lift it. Lifting it the hard way seems will be easier for me to understand.

 

[zoobyshoe]

uh, I'm so confused still. sorry to be a burden with my apparent stupidity (unless what I'm trying to 'get' isn't all that simple.i might take an iq test when i figure this out to hopefully find I'm not as dumb as i feel!) Formulas don't help me much, but I understand the concepts now. It's something i need to 'get' out on my own, but hoping it can be guided. this isn't a big joke, i don't get it still.
I'm drawing a blank on a simple experament:
A large tupperware box & a container of cat litter tied to the end of a 7 foot pole.
Putting the pole on the tuperware and sliding the pole to and fro.
-when the distances of my hand and the litter to the box are equal, I'm pressing as hard as the litter weighs. that's very easy to understand.
-when the litter's closer to the box than my hand, it's easier to hold.
-When my hand's close to the box and the litter is far, I have to press down like 10X the weight of the litter to lift it. Lifting it the hard way seems will be easier for me to understand.

Sounds to me like your experimentation exactly agrees with the formula.

 

[mechaadv43]

i meant don't understand the formulas.

 

[zoobyshoe]

i meant don't understand the formulas.

Do you want to?

 

[eapst28]

The same amount of work is done. Displacement of the bar at the far end is, let's say 3x, and displacement at the work end is only x. Therefore you can move more mass on the "x" end and the same amount of work is done at both sides.

 

[mechaadv43]

ha, i finally get it. doing it the hard way helped. if the weight is say 6 feet from the fulcrum and my end only has to be pushed down 4 inches to make it level, then all the energy needed for the weight to go up about a foot and a half is = to the energy i need to push my end down only 4 inches, thus I push harder, but for less distance. The fact that advantage end moves a larger arch/distance was almost unbelievable with only a 7 foot pole, it's harder for me to realize it, it's easier for me to see when I took the spot right past the pivot.

 

[zoobyshoe]

ha, i finally get it. doing it the hard way helped. if the weight is say 6 feet from the fulcrum and my end only has to be pushed down 4 inches to make it level, then all the energy needed for the weight to go up about a foot and a half is = to the energy i need to push my end down only 4 inches, thus I push harder, but for less distance. The fact that advantage end moves a larger arch/distance was almost unbelievable with only a 7 foot pole, it's harder for me to realize it, it's easier for me to see when I took the spot right past the pivot.

Excellent! You see the trade off in distance and force!

Technically what you're pushing down with is not 'energy' but force. The weight on the other side is also applying force. Getting on the wrong side allowed you to actually feel how much force the other side exerted being 6 feet from the fulcrum. That was a good idea to try.

 

[mechaadv43]

oh, i thought it was considerd 'energy' only because i brought the weight to a level standpoint.
Anyway, using no weight at all was also simpler.just took a broom stick and put all but 3 inches over a pivot that was high off the floor, then pushed down on the short 3 inch side only about 2 inches and the other side went up like 4 feet to make it level. Another thing, if i put a lever under a couch sticking under 2 inches from the pivot but my lever end is 4 feet from the lever, i can make the couch go up about an inch very easily, but i in no way lifted the couche's full weright untiill it's totally off the ground, i just used the same force it would take to lift the couch an inch, plus pushing down on a lever is easier than bending over and pulling up on the couch. it's elementary to me now, just wasn't thinking straight. TY

 

[zoobyshoe]

I think the moral of the story is, "Mechanics is something you should try at home."