A few day’s ago I made a blog post whimsically names “Wait a moment…" If you haven’t read it and plan to understand what I’m about to rattle on about, then I suggest you give it a peruse.
For those who have already read it, I have returned to it to correct the ‘middle-of-the-night-writing’, the content that time and time again proves I shouldn't blog after 10pm and expect it to be literary gold. for this reason it might be helpful if you got lost on any particular bit to skim/scan through it again.
At the end of the previous saga I was left with a mental conundrum of the following ilk. I was confused about how the following to statements could both be true and yet be consistent with each other.
1: A tower, lying on one of its long edges is picked up from one end using a 1:1 pulley system. The lifting direction is kept at all times perpendicular to the floor. If, at any point the system is stationary (ie, the tower is held at a particular angle to the floor) the force applied at the lifting point will be the same as at any other angle the tower might be held at.
2: When the tower is balancing on it’s end the the force applied at the lifting point is simply the force applied down on the other end of the pulley system. this could range from 0N to mgN where m is the mass of the tower and g=9.81m/s.
I struggled with the idea of these two being consistent as I struggled to see what was happening as you moved from one state to another. If you are counter-weighting a tower up onto its end, at which point would a newton meter at the lifting point stop weighing the tower and start weighing the person counter-weighting the tower (which we can assume has less mass.)
The key here is appreciating that they are completely different states, one is stable without any force needed (ie when the tower is on it’s end) and the other state requires an external lifting force to keep it in a stable condition (when the tower makes an angle to the floor that is not 0 or 90 degrees).
The angle at which the tower moves between these two states depends on the dimensions of the tower and is simply the point at which the tower’s center of mass moves past the fulcrum. The other consideration is where on the end of the tower the lifting point is as this will make a difference to the stable points. The following diagram should help explain where these stable points are.
The a,b,c labels on the upright tower show 3 possible lifting points and the diagram shows the 3 stable positions for lifting point 'a’. the first of course is before you start lifting. In this positions and lifting force of between 0 and m/2 will result in no movement from the tower and any fore that exceeds that will cause the tower to start pivoting around 'F’, the fulcrum.
The next stable point is when the center of mass of the tower (which I will assume to be in the middle) is directly above the fulcrum. At this point a lifting force up to m will not cause the tower to move, any force above this will cause it to rise off the floor. While I am calling this state stable, if no lifting force is applied, the tower would probably fall over, with either direction equally likely.
The final stable point is when the tower arrives on its end, which reacts in the same way to the first stable point.
Lifting from points b and c have the first and last stable points just the same as b however the second stable point will occur when the following expression is satisfied;
Where x and y are the horizontal distances between the line perpendicular to the floor and the fulcrum and the lifting point and center of mass respectively.
So, there it is.
p.s. None of this would work quite as I described it in reality because of my assumption that the center of mass is at the center of the tower, which it definitely isn’t. Don’t you love real life…
p.p.s One tower is a little over 300kg and we lifted two of them…
p.p.p.s Sorry for using weight and mass interchangeably this is a side effect of the term ’counter-weighting’.
p.p.p.p.s Anyone paying attention might have spotted that keeping the lifting pointing always pulling perpendicularity to the floor would be tricky. In my head I imagined that as happening by having the top pulley on a zip line type set up controlled most likely by a separate pulley system which would allow the pulley to be slid back and forwards in a straight line. I'm ignoring the associated issue of trying to hang over 600kg (generating over 1.2T when lifted) off what is effectively a tight wire…
p.p.p.p.s - I confused two ideas here, one of lifting two towers which doesn’t require a pulley on a zip line and one of tipping one tower up on its edge. The later which would benefit from a zip line would generate a force of mg/2 on the zip line. This is because there would be two top pulleys which would share the load,of which one would be on the zip line. This diagram should clear it up. All labels give an indication of force involved. To get force in Newtons, multiply quantity by g=9.81m/s.
So, for one tower at 300kg, that would be just under 1500N, the equivilent of 3 fairly small people, fairly achievable. Finally, it did occur to me that the zip line would not need to be under tension and could hang loose, while this would stop it running smoothly, causing the left hand counter-weighter to have to work harder. This would make the whole thing a little more manageable.