Balsa Man wrote:... a little bit of taper gives you a little bit of latitude (for bucket swing, for things not being "true." More taper gives you more latitude. Thoughts on this?
Thanks
My take on this issue:
When we are considering the over-turning of a tower due to the swinging of the sand bucket, there are only three factors (quantities) that we need to worry about, the shape of the chimney is not one of them. The relevant quantities are: the height of the tower, the distance between the support points, and the angle that the sand bucket makes with the horizontal axis.
The following diagram shows a generic tower. You can assume any shape for it as long as:
(1) It has only two support points (we are using a two-dimensional model here) and
(2) The load chain is attached to the top of the tower and passes through the support points at the base.
Let H denote the height of the tower and W be the distance between the support points. Also, let’s replace the bucket and the chain with a downward force applied at the top of the tower, as shown below.
In the above diagram, the support points are labeled A and B, and the point of application of the load (at the top of the tower) is labeled C.
According to Newton’s third law, for every action, there is an equal and opposite reaction. Therefore, since the tower is exerting a force of P on the table, the table reacts by applying an opposite force of P to the tower. This means the table exerts a force of P/2 on the tower at each support point, as shown below.
Since the algebraic sum of the forces applied to the tower (a downward P and two upward P/2s) is zero, we say the tower is in equilibrium, no tipping over yet.
As the bucket starts to move, the direction of P changes, as shown below.
This causes a change in the balance of the two reaction forces at A and B. If the bucket moves to the right (the scenario shown above), then the reaction at B increases while the reaction at A decreases. This eventually leads to the reaction force at A to vanish, as depicted below.
This happens exactly when the line of action of the applied force (P) passes through point B. Past this point, any additional increase in the angle of swing would cause the tower to tip over. Therefore, the critical angle of swing, the angle that makes the chain pass through point B, reduces the reaction at A to zero, and initiate the tipping over of the tower can be easily calculated using simple geometry. The angle equals to the inverse tangent of (W/ 2H). Here is a numerical example.
Example : A tower is 70 cm tall and 10 cm wide (the distance between the base of the legs). If the sand bucket is carrying an unknown amount of sand, how much (in terms of degrees) it can swing in one direction before the tower tips over?
H= 70, W = 10
Tangent of Critical Angle = (W/2H) = ( 10 / 140 ) = (1/14)
Therefore, Critical Angle = 4.09 degrees.
Note that this calculation does not involve any information about the shape of the chimney other than its height.
Having said all of this, however, there are two practical factors that make this kind of instability unlikely for tall towers.
1) The legs generally are not strong enough to carry the extra load being shifted to them as the bucket swings in one direction. A leg failure will take place before the tower tips over.
2) The narrow width of the chimney limits the amount of movement of the chain. It is unlikely that the angle of swing would reach its critical value.
