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

[mechadv44]

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

 

[tiny-tim]

welcome to pf!

hi mechadv44! welcome to pf!

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

no, the energy is the same…

you can't get extra energy for nothing

only the force is less

 

[mechadv43]

same person who started thread, different name, sorry password problem..
I meant force though, can it be explained in plain english as opposed to just a formula why being further uses less force?thanks

 

[waynexk8]

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

Clue.

The crow bar end you are pushing on, goes further than the other end.

Wayne

 

[tiny-tim]

same person who started thread, different name, sorry password problem..

goodness! are there 44 of you?

I meant force though, can it be explained in plain english as opposed to just a formula why being further uses less force?thanks

force times distance equals work done (= energy supplied) …

if the two ends of the lever are at different distances from the fulcrum, so that they move different distances when the lever turns, then different amounts of force will do the same work, and will supply the same energy

 

[mechadv43]

I said that i already know that having a longer lever on the force side of the folcrum requires less force. what I'm asking is WHY WHY WHY does it make it easier? why is a longer lever so special if still moving the same object!?Is this just something that exists and can't be explained?

 

[Lsos]

I think I asked this question once, too. The answer I got was essentially that yes, that's just the way the universe works. Fortunately it's a very simple thing and you'll see manifestations of it everywhere in everyday life. It quickly starts to just "make sense".

Climbing stairs for example is easier than going straight up a wall. A door is easier to open if the handle is on the far side of the hinge. A piece of wood is easier to split when hit with the narrow end of an axe, a screw is easy to screw in and yet it holds things together very tightly, gears, levers, pulleys, even a winding road up a mountain...all take advantage of the fact that work is force x distance, and all these mechanisms simply trade distance for more force.

Unfortunately there are also a lot of concepts in physics where the answer is also "that's just the way it is", but they DON'T make any sense. You have to go in pretty deep to start getting to these concepts, though.

 

[russ_watters]

I said that i already know that having a longer lever on the force side of the folcrum requires less force. what I'm asking is WHY WHY WHY does it make it easier? why is a longer lever so special if still moving the same object!?Is this just something that exists and can't be explained?

Because conservation of energy requires it isn't a good enough answer?

 

[Michael C]

I said that i already know that having a longer lever on the force side of the folcrum requires less force. what I'm asking is WHY WHY WHY does it make it easier? why is a longer lever so special if still moving the same object!?Is this just something that exists and can't be explained?

"WHY" questions are very hard to answer. Feynman explains just how hard it is. When asked why magnets attract each other, he explains that he can't give a simple answer. If you prefer reading a text instead of watching a video, here's a transcript.

For your particular "why" question, maybe Archimedes can help.

 

[sankalpmittal]

I said that i already know that having a longer lever on the force side of the fulcrum requires less force. what I'm asking is WHY WHY WHY does it make it easier? why is a longer lever so special if still moving the same object!?Is this just something that exists and can't be explained?

The most important concept is (as already typed by russ_watters in previous post) law of conservation of energy.

This law states that total energy in the system remains constant , i.e. energy can neither be created nor be destroyed.

Effort E Newtons upward
|​

<-------------------------------------|-------------------------------------------------------->
|

Fulcrum​

Load​

(Axis of rotation)​

(L Newtons downwards)​

Now I am not taking in account load arm and effort arm because lever is not in equilibrium.
Here L >>>> E , so by conservation of energy :
Work Input = Work output
L*d = E*x
(* is multiplication )
L/E = x/d
since L>>>>E , so x>>>>d

x arc >>>>>> d arc covered in circular path with Fulcrum as axis of rotation here.

So here
L* d arc = E*x arc

As work = force * displacement

By conservation of energy we get this result.

Note : We talk about like
Load * Load arm = Effort * Effort arm , when lever is in equilibrium. Longer the effort arm , more the mechanical advantage. This is principle of lever.
Obviously think logically that one guy sitting at longer distance apply force less to counterbalance because distance being more will dominate the counter force effect !Its centre of gravity will be displaced and will be posed with less friction...
(And please we neglect friction ! ) Yes ! Centre of gravity will dominate at the side of the guy sitting at longer distance.

 

[HallsofIvy]

Let's say you have a lever of length 6 m and the pivot is 2 m from the object being moved so that your end of the lever is 4 m from the pivot. As you move the lever through angle \theta radians, you will have moved your end of the lever a distance 4\theta meters. At the same time the object will have moved a distance of 2\theta meters. Since work (energy) is "force times distance" and energy is conserved, the work you do, the force you apply to the lever times 4\theta must be equal to the force applied to the object by the lever times 2\theta:
F_{object}(2\theta)= F_{you}(4\theta)
which reduces to
F_{object}= 2F_{you}
Showing a mechanical advantage of 2 to 1- the force applied to the object is twice the force you apply to the lever.

 

[Studiot]

I'm suprised no one has mentioned the link from mechanical advantage to velocity ratio and mechanical efficiency.

 

[jambaugh]

Here's a way to understand mechanical advantage in general terms. Energy is always conserved. [edit: as mentioned by Tiny-Tim]

When using any kind of pulleys or gears or levers, the power transferred through (work per unit time) is constant and so the work done over an interval is constant.

So, for example if you have a lever, applying a force F on one side and moving a distance \Delta x, it yields an amount of work \Delta W = F\Delta x
That must be equal to the work done on the other side of the lever so you get:
F_1 \Delta x_1 = F_2 \Delta x_2
You can thus use mechanical advantage to double the force but it will halve the motion.
(classic lever with fulcrum 2/3 at 2/3 the length so the two sides has a 2/3 : 1/3 or 2:1 ratio.

Apply 10 Newtons to one side and you'll get 20 Newtons on the other... however move the one side 5 cm and the other side will only move 2.5 cm.

10*5 = 20 * 2.5 = 50 Newton cm = .5 Newton m = 0.5 joules.

You can also generalize to pressures and change of volume torques and angles of rotation and also to voltage and flow of charge.

Work = force times \Deltadistance = pressure times \Deltavolume = torque times \Deltaradian angle = voltage
\Deltacharge
likewise time rate of change of work which is power is:
power = force times speed= pressure times volumetric flow rate = torque times angular speed = voltage times current.

This assumes the ideal case with 100% transmission efficiency. Friction and other loss effects will suck up some of the energy or power in an actual example.

So assume you want to lift an engine block which masses 1/2 a metric ton=500kg.
It thus weighs 500kg x 9.8m/s^2 = 500 x 9.8=4900 Newtons. (Multiply mass times gravitational acceleration to get weight as a force)

You wish to lift it 2 meters. That means you want to do 9800 Joules of work. (Joule = Newton Meter)

You can do this with a 5:1 pulley system where you pull on the rope with a 100kg lifting force (980 Newtons=1/5 the force) by pulling with that force 5 times as far (10 meters).

You can also do this with a pneumatic lift applying say 1 torr = (100,000 Pa = 100,000N/m^2 = 100,000 Joules / meter^3)
by displacing .098 cubic meters= 98000cc's.
(or use 100 torr displacing 980cc's that's a bit under 1 atmosphere pressure)

Or use a 240V electric wench which will require 9800/240 ~ 40.83Coulombs. (at 10Amps that will lift it in 4.083 seconds.)

Mechanical advantage is a matter of spreading the work out over a greater amount of motion so it requires less force. In the end though work input = work output.

Archimedes said “Give me a place to stand and with a lever I will move the whole world.” what he neglected to say was “but not very far ; ) ”

 

[mechadv43]

Sorry if the answer’s already posted and I’m just not bright enough to see it. But, what I’m really asking is can a person (the smartest person ever for argument’s sake) move a lever with mechanical advantage and actually understand/feel it in their mind how it‘s requiring less force, or is it ‘just how the universe works?’ Is it how sankalpmittal said, center of gravity will dominated at the longer side of the lever?

thanks

 

[sankalpmittal]

Sorry if the answer’s already posted and I’m just not bright enough to see it. But, what I’m really asking is can a person (the smartest person ever for argument’s sake) move a lever with mechanical advantage and actually understand/feel it in their mind how it‘s requiring less force, or is it ‘just how the universe works?’ Is it how sankalpmittal said, center of gravity will dominated at the longer side of the lever?

thanks

See the post of Michael C i.e. post 9 , it has link to a good site which has your answer I think.

Here is my answer :
I think you are confusing principle of lever with law of conservation of energy.

http://www.astarmathsandphysics.com/o_level_physics_notes/o_level_physics_notes_moments_the_lever_principle_html_210ae596.gif

Now see , in above image that lever is balanced at fulcrum with distance d > distance D
Now since lever is balanced , then anticlockwise moments equals clockwise moments.

Then ,
w*d = W*D
since w<W and d>D
Then
W/w = d/D
Now see that resultant force will act on the point which divides the ratio of distance "d+D" in d:D.
As we know that d> D so the resultant force will dominate at the side of distance d.
Hint : Here centre of gravity will be at midpoint of distance "d+D" that is at the side left of fulcrum in this system ! Also think about section formula of coordinate geometry.
W/w = mechanical advantage as well
M.A. = d/D
Hence the mechanical advantage will be with the small stone of weight w.

Does this help ?

Edit : Tiny-Tim thought that you were asking about law of conservation of energy in which work input equals work output. That's different matter although here it applies as its a universal law. Halls Of Ivy too answered your question in different way. jambaugh too had a great explanation.

 

[jambaugh]

Sorry if the answer’s already posted and I’m just not bright enough to see it. But, what I’m really asking is can a person (the smartest person ever for argument’s sake) move a lever with mechanical advantage and actually understand/feel it in their mind how it‘s requiring less force, or is it ‘just how the universe works?’ Is it how sankalpmittal said, center of gravity will dominated at the longer side of the lever?
thanks

Meditate on the idea that Force is a Rate of work per distance. Say it out loud, "Force is work per distance!"

As I move a lever I am doing a given amount of work, since the lever transforms distance traveled it "dilutes" or "concentrates" the work per distance i.e. the force.

 

[sankalpmittal]

Refer this image : http://postimage.org/image/51wfdnhwn/

Mathematical reason :

Kinetic energy of load arm = Kinetic energy of effort arm
By this we get :\
Mass of load/Mass of effort =Inverse Ratio of square of velocities
By image we get
Mass of load = 4 times mass of effort
As velocity of effort is twice the velocity of load
So Velocity ratio = 2
For an ideal machine
Mechanical advantage = Velocity ratio
So
Load = twice of effort
From this we can get that acceleration in effort is twice of load.

Also power input = power output
or work input equals work output
So greater the arm less is required force for lever to be in equilibrium.

Theoretical reason : Since angle θ is same in both triangles (see image) and base of effort side is double of base of load side then its obvious that effort arm will cover greater arc distance than load arm. Since work input equals work output , we have to say that effort must be half of load for same work distance. Less mass more acceleration. Also note arc distance by effort is twice of that of load ,as ratio has to be maintained. Also note that initially both arms are at rest.

 

[JHamm]

I think the OP is asking, not for a way to prove that levers supply a mechanical advantage, but for how the atoms in a lever interact so that the forces at one end are different to the forces at the other.
i.e. what is the cause of mechanical advantage?

 

[jambaugh]

I think the OP is asking, not for a way to prove that levers supply a mechanical advantage, but for how the atoms in a lever interact so that the forces at one end are different to the forces at the other.
i.e. what is the cause of mechanical advantage?

Ahhhh, yes that's a different question. For that it is important to understand (if I'm lifting you with an off center see-saw) that it is not my downward force which lifts you but rather the force the fulcrum is pushing upward to both lift you and oppose my downward force.

It is the same force that keeps us from falling to the center of the Earth though gravity is trying to make us do just that.

It is also an important consideration when setting up a mechanical advantage system. You can have a strong lever and good advantage but if you use a weak fulcrum it will break before you lift your load.

Ultimately to get a full understanding I think one should study vectors and understand how the various directed forces add up and cancel.

 

[sophiecentaur]

I think the OP is asking, not for a way to prove that levers supply a mechanical advantage, but for how the atoms in a lever interact so that the forces at one end are different to the forces at the other.
i.e. what is the cause of mechanical advantage?

I think that is taking things much too far in one go. All in one breath you're supposed to have a grasp of inter-atomic forces / quantum mechanics AND the Principle of Moments. Moments are soooo much easier to cotton on to that they are taught to 13yr old kids. QM is seldom grasped until Degree level.

'Looking back' one can often see a link between elementary and advanced stuff but only when 'you know the answer anyway'.

Also, I am not at all happy about the term "Kinetic energy of the Load arm" in the first post. Moments have nothing at all to do with MASS or Kinetic energy. They relate FORCES and DISTANCES. If you consider Power, then you should be discussing Force times Speed - which is measured in Watts, in the end.

 

[JHamm]

I think that is taking things much too far in one go. All in one breath you're supposed to have a grasp of inter-atomic forces / quantum mechanics AND the Principle of Moments. Moments are soooo much easier to cotton on to that they are taught to 13yr old kids. QM is seldom grasped until Degree level.

I agree the question is somewhat innapropriate, it's just what I think he was trying to ask us.

 

[zoobyshoe]

Sorry if the answer’s already posted and I’m just not bright enough to see it. But, what I’m really asking is can a person (the smartest person ever for argument’s sake) move a lever with mechanical advantage and actually understand/feel it in their mind how it‘s requiring less force, or is it ‘just how the universe works?’ Is it how sankalpmittal said, center of gravity will dominated at the longer side of the lever?

This might help: if you can understand how a magnifying glass takes the harmless light from some given area and makes a very small, intense, not so harmless, hot spot that can start a fire, then mechanical advantage is the same principle. The magnifying glass takes light from a large area and concentrates it into a very small area. Likewise, a lever takes the small force which is "spread out" over a long distance and concentrates it into a much smaller, but more forceful working distance. The "hot spot" created by a magnifying glass only exists in a much reduced area, the increased force of the lever only acts over a much reduced distance. Rather than a "force multiplier" think of it as a "force concentrator", because that's exactly, and more precisely, what it's doing. You are taking something that is "spread out" and concentrating it. This is why the input = the output, and there's no violation of any conservation laws. All you're doing is concentrating what you have.

I think anyone who has ever jacked up a car ought to be able to form the intuitive understanding you ask about. You are doing all this winding of the crank (with a screw jack) covering much distance with your hand, while the car, conversely, moves upward very, very slowly by comparison. The jack is "collecting" all the force you apply and slowly concentrating it to raise the car. With each turn of the crank you apply a certain force over, say, 3.1416 feet (if the crank circle radius is 6 inches, say) and the jack concentrates all that force that was spread out over all that distance to something compact that actually raises the car about 1/8 of an inch.

 

[sankalpmittal]

Also, I am not at all happy about the term "Kinetic energy of the Load arm" in the first post. Moments have nothing at all to do with MASS or Kinetic energy. They relate FORCES and DISTANCES. If you consider Power, then you should be discussing Force times Speed - which is measured in Watts, in the end.

Are you referring to my post , post 17 ?
Why do you think that moments has nothing to do with kinetic energy ? Kinetic energy is the energy possessed by a body or an object by virtue of it being in motion. When we apply force on effort arm on one side , then we are applying muscular energy to it. Aren't we ? Energy is conserved and so that energy sets the effort arm in downward motion , say. Being in motion it has to possesses kinetic energy. This in turn sets load arm in upward motion , and so it also must possesses kinetic energy equal to effort arm due to law of conservation of energy which is a universal law.

Moreover , whenever we do work , we say or we mean in Newtonian physics , that if an only if that object covers some displacement. Hence there is momentum in that object and being in motion it must have kinetic energy.

Moment of an arm is the turning effect of that arm also known as torque. It is force times perpendicular distance of line of action of force from point of rotation.
So its different from the force. Nonetheless , if there is moment (clockwise or anticlockwise) , there is kinetic energy. Right ? Please correct if I am wrong..

__________________________________________________________________

So OP is asking about interaction of molecules ? 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.

But we assume the lever weight massless in practical applications and in theories of textbooks and in studying laws like law of lever by Archimedes. Do we not ?

 

[sophiecentaur]

In my opinion, if you are trying to help someone to understand something about Physics then we are duty bound to use the correct and well defined terms. If we don't then we can confuse them and leave them worse off than they were in the first place.

You want to talk about "kinetic energy of a force"? Well, perhaps you could quote a reputable source in which the term is used. That would be the acid test. I certainly never came across such confusion in a textbook.

A lever doesn't have to be moving for the force to be 'multiplied'. No Mass is mentioned in the Principle of Moments. I know there is a method for working out forces in structures that is referred to as 'virtual work', in which you allow the structure to distort by an infinitessimal amount and then discuss the 'force times distance' involved. That does NOT involve the movement of masses (KE).

You may have a 'feeling' about this and this may be your personal way to get closer to the subject but please don't assume to be enough of an authority about it to instruct others in this approach. It is basically flawed because it mixes dimensions in a very dodgy way.

I will withdraw my objection if you can give me a credible reference, though.

 

[sankalpmittal]

In my opinion, if you are trying to help someone to understand something about Physics then we are duty bound to use the correct and well defined terms. If we don't then we can confuse them and leave them worse off than they were in the first place.

You want to talk about "kinetic energy of a force"? Well, perhaps you could quote a reputable source in which the term is used. That would be the acid test. I certainly never came across such confusion in a textbook.

A lever doesn't have to be moving for the force to be 'multiplied'. No Mass is mentioned in the Principle of Moments. I know there is a method for working out forces in structures that is referred to as 'virtual work', in which you allow the structure to distort by an infinitessimal amount and then discuss the 'force times distance' involved. That does NOT involve the movement of masses (KE).

You may have a 'feeling' about this and this may be your personal way to get closer to the subject but please don't assume to be enough of an authority about it to instruct others in this approach. It is basically flawed because it mixes dimensions in a very dodgy way.

I will withdraw my objection if you can give me a credible reference, though.

Sir , my textbook says that we can always correlate work and energy which is given in work-energy theorem. Also , my textbook says that whenever there is motion there is kinetic energy and whenever there is momentum (angular or linear) , there is kinetic energy. Work done is force times displacement covered in the direction of it. Momentum is motion possessed by that body. Hence body will have momentum when we do work on it and thereby it will have kinetic energy.

Talking about moments or torque it is force times perpendicular distance of line of action of force from point of axis. Of course it has same unit as that of work which is Newton metres in SI system. We also say that
1 Newton metre = 1 Joule
But we cannot say that work and moments are same. Thay are entirely different.
Yet when lever arm moves downward , say and other arm moves upward , then lever as whole suffers clockwise couple. Yet there is motion in it and so kinetic energy.
Is there not ?

Sir , please tell me what's wrong in my previous post , post 23 and tell me where that analogy fails.
Thanks...

 

[sophiecentaur]

. . . . .
Sir , please tell me what's wrong in my previous post , post 23 and tell me where that analogy fails.
Thanks...

I think the main problem is that you have Implied a connection between Force and Mass which is not there and used that to draw a false conclusion.

Of course there is often an 'association' between Force and Kinetic Energy in machines of all sorts. However, there is more to Physics than mere associations. We try to do better than that.

Firstly, I notice that you haven't quoted a reputable reference for the expression "Kinetic energy of a Force" and I am not surprised. The textbook you quote from states the 'real' relationships and does not even imply that 'your' phrase is valid. You have constructed your own argument by extending too far from what you have read in the book.

One simple way to demonstrate your inappropriate use of KE in the context of levers is to consider a simple 2:1 length lever with balanced masses (Ratio 1:2). If you allow some movement (say the large mass moves downwards) and do some simple calculation for the KE of each mass, you find that the smaller mass will have TWICE the KE of the larger mass. The velocity will be twice and the mass is half, so the mv²/2 is NOT the same in each case. So, although we have done a calculation involving this lever, it really doesn't lead to any useful conclusion and definitely not an equation that is of any use for solving a 'balance' problem. There is some other form of Energy which would need to be considered as well.

Another nail in the coffin is to consider Friction. When you turn a wrench against a very sticky thread, your hand / body may have some kinetic energy but what is moving on the other implied end of the lever? A nut, with a mass of just a few grams, rotating very slowly. Any energy that you may be putting into the system is not turning up as identifiable Kinetic Energy (there will be some KE in the form of internal energy - heat - but this is outside of your analysis and doesn't count). Work has been done but KE is not relevant - or only a tiny part of the situation.

 

[zoobyshoe]

One simple way to demonstrate your inappropriate use of KE in the context of levers is to consider a simple 2:1 length lever with balanced masses (Ratio 1:2). If you allow some movement (say the large mass moves downwards) and do some simple calculation for the KE of each mass, you find that the smaller mass will have TWICE the KE of the larger mass. The velocity will be twice and the mass is half, so the mv²/2 is NOT the same in each case. So, although we have done a calculation involving this lever, it really doesn't lead to any useful conclusion and definitely not an equation that is of any use for solving a 'balance' problem.

I am glad you pointed this out because I tried it and you're right. The fact the lever arms have kinetic energy seems, therefore, to be completely beside the point. Bringing it into the mix does not serve to clarify anything.

 

[sophiecentaur]

I am glad you pointed this out because I tried it and you're right. The fact the lever arms have kinetic energy seems, therefore, to be completely beside the point. Bringing it into the mix does not serve to clarify anything.

Glad someone took my point. Not only does introducing KE miss the point but it's a potentially seriously confusing thing to introduce.
This is a great example in which someone's own visualisation of a situation, which may work fairly well for them, personally, in many circs should not be used as a tool for 'explaining things' unless it is 'rock solid'.
If anyone takes the huge step of 'explaining' something on a forum like this, they are taking on the mantle of TEACHER. God knows, there are enough professional teachers who actually struggle with many of the basics that they are expected to teach and what they have told students can be to blame for a lifetime of subsequent conceptual problems.

 

[jambaugh]

Moment of an arm is the turning effect of that arm also known as torque. It is force times perpendicular distance of line of action of force from point of rotation.
So its different from the force. Nonetheless , if there is moment (clockwise or anticlockwise) , there is kinetic energy. Right ? Please correct if I am wrong..

On this, slightly. The force moment is as you say a torque a generalized force. It does not imply kinetic energy when there is no rotational motion. One note as to why torque and energy have the same units, since a generalized force is work done per generalized motion if the units of that motion are pure numbers (as with radians) then the units of force will equate to units of energy. Technically though there is a distinction. As said, force is not work. The proper units of torque would be e.g. Newton-meters per radian = Joules per radian.

If one wants to get super technical, mathematically a torque (or other gen. force) differs from work in that it is a linear operator mapping differentials in one coordinate to differentials in another not a stand alone quantity. You see that in the way force is measured, you have to do a little bit of work over a little bit of displacement to see the ratio. Look explicitly at how a scale or a torque meter works. (or a volt meter, or a pressure meter...)

 

[sophiecentaur]

On this, slightly. The force moment is as you say a torque a generalized force. It does not imply kinetic energy when there is no rotational motion. One note as to why torque and energy have the same units, since a generalized force is work done per generalized motion if the units of that motion are pure numbers (as with radians) then the units of force will equate to units of energy. Technically though there is a distinction. As said, force is not work. The proper units of torque would be e.g. Newton-meters per radian = Joules per radian.

In the dreaded Imperial system, this is taken care of by using foot pounds for work and pounds feet for torque. That could be the one single way in which Imperial is better!