Re: [Phys-L] A force multiplier

[John Denker <jsd@av8n.com>]

On 02/16/2013 08:54 PM, Ludwik Kowalski wrote:
> Suppose the maximum force a person can pull is F1= 200 lb while the
> force needed to move the car from the mud is F2=4000 lb. The angle
> alpha, needed, when F2/F1=20, according to the above formula, is
> about 1.4 degrees. Suppose the original horizontal distance from the
> bumper to the middle of the road is 10 meters, and that the rope is
> originally not stretched. The man pulls it up. The needed 1.4 degree
> is reached when the middle of the rope is only 0.24 meters above the
> road. The distance to the car (along the rope) changes from 10 meters
> to 10.24 meters. Such 2.4 % change, in the length of the rope, is not
> large enough to generate the needed F2/F1 ratio. The "spring
> constant" of a typical rope, in other words, is not large enough to
> satisfy the theoretical formula. That formula, however, would be
> reliable if the rope were replaced by a thick steel cable, as
> indicated by John. D.

When I run the numbers, I get an even more discouraging result.

The origin of the mechanical advantage lies in the geometry and
trigonometry of the situation.  The transverse deflection is
proportional to sin(theta), while the longitudinal motion is
proportional to 1-cos(theta).  For small theta, the former is 
first order, while the latter is second order!  You can't use
simple proportional reasoning.  (Maybe that's the answer to the
original question as to why this mechanism is counterintuitive
in the small-angle case.)

rope length	20	m			
F2      	4000	lbf		
F1      	200	lbf		
					
ratio    	0.05				
half    	0.025				
					
theta   	0.0250   	radian	i.e.	1.43	degree
transverse 	0.25    	m		(agrees with LK)	
deflection
				
cos theta	0.999687				
1-cos theta	0.000313		i.e.	0.03%	
motion of end	0.006251	m	i.e	6.25	 mm

In other words, I calculate that even with an infinitely non-stretchy
rope, the car moves only 6.25 mm  (which is 40 times less than the LK
result).

Even with a steel cable 1 cm in diameter, the putative motion is an order
of magnitude less than the stretch in the cable.

modulus of 	65000000000	Pa	i.e.	0.64	megabar
wire rope
diameter	0.01     	m			
area    	7.854E-05	m^2			
force-scale	5105088.062	N			
relative exten	0.0035				
total stretch	0.0698	        m   	i.e	69.81	mm

The spreadsheet I used to calculate all this is at
  http://www.av8n.com/physics/cable-mech-adv.xls

==================

It is amusing to reflect on what works and what doesn't:
 -- Dimensional analysis fails in this case.  Proportional
  reasoning fails.
 -- Non-dimensional scaling analysis works fine.  You have to
  work out the physics in enough detail to realize that the
  longitudinal effect is *second order* in the transverse effect
  (for small theta).

Take-home lesson:  Dimensional analysis is a poor man's approximation
to a scaling analysis.
  http://www.av8n.com/physics/dimensional-analysis.htm
  http://www.av8n.com/physics/scaling.htm