How a force is amplified in a lever from the standpoint of internal stresses?

[fog37]

Hello,
While trying to understand how a lever truly works and a force can be amplified using a larger lever arm, I read a thread on how levers amplify forces (https://physics.stackexchange.com/questions/22944/how-do-levers-amplify-forces ) and the discussion involves stresses, internal torques and internal forces.

Let's consider the static situation of a rigid bar with an off-center pivot under it. There are two different lever arms $L_1$ and $L_2$ with $L_2 < L_1$. Two unequal forces $F_1$ and $F_2$ are applied to the ends of the bar and the two opposite torques of equal in magnitude keep the bar from rotating. The forces are $F_2>F_1$ generate internal stresses inside the rigid body that are distributed along the bar.
Aside from understanding the simple equation $L_1 F_1=L_2 F_2$, how can we better understand the fact that $F_2$ is larger than $F_1$? The parts of the bar that are not at the end of the bar under under stress. The two external forces generate internal local torques at different points along the bar. I still don't understand how a larger $F_2$ force arises when the other force $F_1$ is smaller.

How an applied force gets amplified on the other end from the standpoint of internal stresses?

Thank you!

 

[Halc]

While there are internal stresses, the amplification of force comes from balanced torque about the fulcrum, and not the internal torques around any other point on the bar which are relevant only if one is worried if the bar might break under the stress. It is the position of the fulcrum which determines the $F_2$ that results from the applied $F_1$.
Nothing is apparently moving, so $F_3+F_2+F_1=0$ meaning the force at the fulcrum also depends on its position along the bar.

I didn't read the other thread so I don't know what all is covered by it.

 

[fog37]

Thanks. I see your point and how the two torques equal each other and the net force is zero. But I am trying to figure out how the larger force arises. When we use a lever and apply a force at one end the larger force at the other end shows up because of some internal transmission of the stress from where the force is applied to the other end...

 

[hutchphd]

The process is not and cannot be strictly internal. Archimedes said "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."
The lever works because it causes the fulcrum to supply the force.

 

[Baluncore]

The lever works because it causes the fulcrum to supply the force.

The fulcrum must not move, so no work is done at the fulcrum, and no energy is lost there.

A lever that doubles a force, halves the distance moved, and halves velocity. The rate work is done, or energy flows is the same at both ends of the lever.

 

[Lnewqban]

Hello,
While trying to understand how a lever truly works and a force can be amplified using a larger lever arm...

How an applied force gets amplified on the other end from the standpoint of internal stresses?

Thank you!

There are three classes of levers.
Please, see:
https://www.engineeringtoolbox.com/levers-d_1304.html

Al three follow the mechanical principle of gaining force at expense of acted distance, as the input energy or work is about the same the output energy.

Win=Wout
Fin x Distancein=Fout x Distanceout

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/lever.html#c1

A force and a distance induce a moment about a fulcrum; and than moment induces a force at certain distance.

Internally, the lever experiences internal forces in forms of shear and axial pairs of forces than change along it.

Please, see:
https://www.engineeringtoolbox.com/stress-d_1395.html

 

[Tom.G]

Well, let's try a 'simple' version that isn't exactly accurate but maybe easier to keep track of.

This got rather lengthy for a word description so it may help you to draw a sketch as we go along.

You may have heard/seen the phrase that 'Energy is conserved.' That is another way of saying that 'what comes out must have come in' from somewhere. Another way of stating it is that 'Energy can be neither created nor destroyed.'

Relating this to a lever with the fulcrum off-center, consider pushing down on the long arm of the lever, perhaps 10 pounds at 5 feet from the fulcrum. You push it down 2 feet.

This gives a torque, (or 'turning force' around the fulcrum) of 50 foot-pounds and a travel of 2 feet.

If the short arm of the lever is 1 foot long it will of course rise 2/5 of a foot. (because the two arm lengths are 1 & 5 feet for a ratio of 1/5, and you pushed the long one down 2 feet, the short arm rises 2×(1/5)= 2/5)

Now we get to that Conservation of Energy mentioned above. The short, 1 foot, arm travels only 2/5 of a foot but the torque is still 50 foot-pounds (that's the 'Energy' that is conserved).

So we have a torque of 50 foot-pounds, divided by the 1 foot short arm length, giving 50/1=50 pounds force.

What happens with a lever is you trade off distance traveled with force. For a given amount of effort the product of the two is the same at each end of the lever.

I hope it helps!
Tom

 

[fog37]

There are three classes of levers.
Please, see:
https://www.engineeringtoolbox.com/levers-d_1304.html

Al three follow the mechanical principle of gaining force at expense of acted distance, as the input energy or work is about the same the output energy.

Win=Wout
Fin x Distancein=Fout x Distanceout

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/lever.html#c1

A force and a distance induce a moment about a fulcrum; and than moment induces a force at certain distance.

Internally, the lever experiences internal forces in forms of shear and axial pairs of forces than change along it.

Please, see:
https://www.engineeringtoolbox.com/stress-d_1395.html

View attachment 292245

Thank you lnewqban. The shear and moment graphs are useful even If I am still trying to understand it.

I see the energy argument (work in = work out) and why the output force must be bigger.

However, it still remains a mystery from an internal stress point of view, how an externa force applied at one end generate a larger external force at the other end of the bar...

 

[Lnewqban]

Thank you lnewqban. The shear and moment graphs are useful even If I am still trying to understand it.

I see the energy argument (work in = work out) and why the output force must be bigger.

However, it still remains a mystery from an internal stress point of view, how an externa force applied at one end generate a larger external force at the other end of the bar...

You are welcome.
Please, read this when you can:
http://alistairstutorials.co.uk/tutorial06.html

If the lever is floating in space, your input force will not result in any output force, only linear acceleration of the center of gravity of the lever and/or angular acceleration of the lever.

In order to have an output force induced by an input force, the lever must be unable to freely move linearly, it can only rotate in one plane about a fixed-to-ground fulcrum (where a reactive force will appear).
The lever is internally rigid and resists deformation; meaning that a external force induces internal forces.

Those internal forces change their magnitudes and are transferred along the length of the lever, and manifest themselves at the output end.
We could loosely say that the combination of the input force and the distance between point of input application and fulcrum becomes a maximum internal moment at the fulcrum, which becomes another combination of output force and the distance between the fulcrum and the point of output application.