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:
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) :
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.
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.