Explain the functioning of the Balancing Bird.
What is the Balancing Bird?
The Balancing Bird is a geometric figure that is similar to a bird, built in a way that when it is supported on its beak it is in equilibrium.
The design and construction of a Balancing Bird obeys to Physics concepts such as the centre of mass and the balance of forces and moments.In the shops we can find many types of Balancing Bird, with varied colours and formats.
The Balancing Bird can serve as a toy and as decoration of homes and gardens. It can also be used as support material in classes of various subjects, to illustrate concepts of Physics and Engineering.
The Balancing Bird will only be in equilibrium, in relation to its beak, if there is a balance of the moments caused by the massof the wings, the body and the tail, with respect to the point where the beak is.
Thus, through the generic expression of the equilibrium of moments;
[pic]g [pic] + [pic]g [pic]g [pic] = [pic] [pic]
Putting g on evidence,
g ([pic] [pic] + [pic] [pic] [pic] ) =g [pic] [pic]
and dividing the whole equation for g, we get;
[pic] [pic] + [pic] [pic] [pic] = [pic] [pic]
Thismeans, [pic] = [pic] [pic]
where, [pic] (1)
On the other hand, considering that the material is homogeneous and of uniform thickness, being m = V e V= AƐ (where Ɛ is the thickness), we can write m= AƐ and substituting in (1) we obtain
[pic] or [pic]
Cancelling [pic]we obtain [pic]
In the same way we can calculate[pic],[pic]
being ([pic] the coordinates of the centre of mass of the system.
Analysis of the functioning of the Balancing Bird
To study the Balancing Bird we thought, initially, as first approach, in a circular disk of homogeneous material and constant thickness whose centre of mass is at its geometrical centre and so, when we support the disk onits centre we can see it in equilibrium, since the sum of the moments in relation to the centre of the disk O will have null sum and for reasons of symmetry and of homogeneity of material.
(Entire disc----> Center of mass in the centre)
If we cut this disc and add some material to get a figure close to the geometry of theBalancing Bird it is natural that the centre of mass of this new geometry (Picture 3) will descend in relation to the position it held previously (Picture 2)
(Cut disc + Rectangle----> the centre of mass descends in relation to the previous situation)
As our interest is that theBalancing Bird stays in equilibrium, when it is supported on its beak, then it is necessary that the centre of mass will occupy the initial position again.
For that we will need to increase the size/mass in the upper zone of the wings.
Simplified Balancing Bird
We drew a simplified figure but which allows explaining how we can design a Balancing Bird.
We considered afigure divided into several rectangular blades with uniform thickness and homogeneous material. This lets you to perform calculations in a simpler way.
We will place the beak of the Balancing Bird at the point of coordinates O(x, y) = (0.4).
As we do not know in advance what should be the dimensions needed to make point O (0.4) the centre of mass, we will make the length of the body/tail of...