What do Leonardo da Vinci (1452-1519), Oliver Hardy (1892-1957) and my former Physics teacher (1926-2009) have in common? I could not expect anyone to know, but am sure this unlikely trio would, if able to view the video included here, be impressed, appalled and amused in equal measure.
So where to begin? In the classic 1932 film The Music Box, Oliver Hardy, along with partner Stan Laurel (1890-1965), attempts to raise a boxed piano from the ground to a first-floor balcony using a block-and-tackle pulley system (Figure 69.1). Even viewers with no knowledge of their legendary incompetence could predict that the consequences involve property damage, widespread débris and sore body parts.
Figure 69.1: The incomparable Oliver Hardy demonstrating his expertise with block-and-tackle
Pulley systems have been employed, safely or otherwise, since the days of Archimedes (287-212 BCE). Put simply: rope sections change the direction of applied force, create a mechanical advantage, and more (load-supporting) ropes yield greater efficiency. The aim is to maximize the ratio of the distance moved by the applied force to that moved by the load itself. Perfect efficiency can never be attained, owing to friction and other energy losses, but a typical 70-80% yield is generally worthwhile.
Da Vinci developed the principles of the pulley system and utilized them in many of his inventions, in particular the revolving (swing) bridge. Further sophisticated applications have adorned the notebooks of engineers for centuries.
My own rudimentary knowledge of pulley mechanics (Figure 69.2) was gained in a British high-school Physics lab.
Figure 69.2: Here, the greater mass, X, will move downward (left), as gravity overcomes the tension in the rope, and come to rest on impact with the ground (right).
Copyright © 2013 University of Wisconsin-Madison
In accordance with Newton’s Second Law of Motion:
(MX × g) – T = MX × a
where MX = mass of object X; g = acceleration due to gravity; T = tensile force exerted by rope; and a = resultant acceleration of X when the system is in motion.
Conversely, Y, being lighter than X, will be pulled upward thus:
T – (MY × g) = MY × a
where MY = mass of object Y; while acknowledging that X and Y will accelerate at the same rate, a, (albeit in opposite directions), and that rope tension, T, is equal in both sections.
If we combine these two equations:
a = g(MX – MY)/( MX + MY)
Next, supposing the mass of X (in order to move downward) is twice that of Y:
a = g/3
That is, X would accelerate downward at one third the rate of gravity. If travelling a vertical distance H, as shown in Figure 69.2, it would impact the ground with a velocity V, where:
V = √(2 × a × H)
For example, from a height of 15 metres, X would hit the ground at 10 metres per second. (This can easily be deduced from first principles, in case you wish to verify my equations.)
So far, so logical; but what if X were a human being and Y a bucket of cement? The motion mechanics would hold true; but, of course, any person hitting the ground at 10 metres per second would risk minor injury. (A skydiver, equipped with a standard parachute, impacts the earth at approximately 7.5 metres per second.) At 20 metres per second or greater, skeletal or muscular trauma would be practically unavoidable. Therefore, in order for a human to escape unscathed from such an adventure, the counterweight, Y, would need to exceed half his bodyweight.
Who, you might be wondering, would be crazy enough to test the veracity of Newtonian mechanics in such a way? Answer: a Chinese construction worker (Figure 69.3). Before playing the video, allow me to explain why I claimed that it is possible to be ‘impressed, appalled and amused’ at the same time. What is impressive is that the guy does not get hurt. He ensures that there is sufficient cement load to curb his rate of descent, insufficient for it to rupture the bucket (which would suddenly increase it), and that he swings clear of the other rope so as to avoid colliding with the loaded bucket on its way up (at exactly the same speed). Appalling sums up both his disregard for personal safety and that the system is in fact woefully inefficient. As for amusing ... well, amusing just is. Happy New Year to all.
Figure 69.3: If my dear Physics teacher were still alive, this footage would probably give the old maverick ideas.
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Copyright © 2014 Paul Spradbery