What is the Mechanical Advantage of this lever? (for pulling tree stumps)

  • #1
I am interested in building a lever system for pulling up small tree stumps. I have seen this demonstrated on youtube videos. For example...
However, I am more interested in calculating the various forces and mechanical advantage for such a system so I can figure out the best lever length, chain strength, and pulling angle. I have yet to find an example calculation for such a system and I'm not even certain which class the lever is. I would assume either class 1 or class 2. Anyway, here is a crude drawing of the system. Any help with identifying class and formula for calculating MA would be greatly appreciated. I would like to know what happens as the lever moves toward the force of pull as it first straightens up then collapsed toward the ground again which also alters the angles during the progress of the pull.

upload_2018-2-28_13-22-18.png
 

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  • #2
stockzahn
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Momentum 1: ##M_g=F_g l sin\theta##
Momentum 2: ##M_p=F_p l cos\theta##

Lever: ##\frac{cos\theta}{sin\theta} = cot\theta >\frac{F_g}{F_p}## ... to pull out the stump
 

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  • #3
sophiecentaur
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formula for calculating MA would be greatly appreciated.
Until you know the details of the construction, you cannot predict what will be the 'Mechanical Advantage' of any machine. MA is the actual Ratio of load force to effort force. Other forces such as friction can reduce the MA and I can see a beam there of which the mass may not be negligible. That beam can be referred to as Dead Weight which may or may to be relevant. What you can calculate from the geometry of the machine is the Velocity Ratio (known also by other names), which is the ratio of distance moved by the effort divided by the the distance moved by the load. This involves either some calculus or just a choice of a 'small' movement or sometimes just the lengths of levers or diameters of wheels. You can see that one angle increases as the other decreases so simple lever lengths won't do the job.
It isn't necessary to actually name the class of lever to work out. the VR The fulcrum is on the ground and that's the best place to take Moments about. There are two angles involved (Note @stockzahn ) because the rope doesn't emerge in a horizontal direction or may not be symmetrical. If we call the angle of the beam to the vertical as θstump, there will be an anticlockwise moment of lengthbeamWeightstump Sin(θstump)
If we call the angle between the rope and the beam as θrope the clockwise Moment will be lengthbeam Tension Sin(θrope) then we can equate those two and solve the equation for the unknown Tension.
It may not be convenient but the maximum moment from the rope is when θrope is 90° which requires it to be higher than horizontal. A second beam can help you here, with a 'lifting triangle', rather than just a simple jib. Look at the design of many cranes and derricks to see what I mean.
The maximum force on the tree will be obtained with the beam just off vertical. But you would not get much movement at that angle so you need to tilt the beam a bit (enough to allow the tree root to be lifted out of the hole.
The weight of the beam also adds an anticlockwise moment (1/2 lengthbeam weightbeam Sin(θstump. Using that could give you an idea of the actual MA. Two beams in a triangle could actually balance each other and remove that moment.
 
  • #4
A.T.
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Momentum 1: ##M_g=F_g l sin\theta##
Momentum 2: ##M_p=F_p l cos\theta##

Lever: ##\frac{cos\theta}{sin\theta} = cot\theta >\frac{F_g}{F_p}## ... to pull out the stump
Note that this assumes the strings are at 90°.
 
  • #5
stockzahn
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Note that this assumes the strings are at 90°.
True, but it vividly shows the idea. Start simple and evolve...
 
  • #6
sophiecentaur
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True, but it vividly shows the idea. Start simple and evolve...
But isn't a practical scenario, is it? How can you have a horizontal rope if the beam is more than chest height? I have had experience of this sort of thing when raising and lowering a 10m sailboat mast. The rope would probably need to be attached to a ground picket if any significant lifting force is needed. This is why I was suggesting the use of two beams and not just one. It's a win win design - pulling at near 90 degrees for the derrick and almost zero degrees for the lifting section.
PS Momentum is Mass times velocity. The Moment of a Force is Force times perpendicular distance.
 
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