Ok,
so 9 months later here are a few corrections, as promised:
Corrections:
(i)
Including the mass of striker
(by just adding it to the mass of the wind sail):
a)
Metal striker: R= 20 mm & t = 8 mm
Ms = r (pr2t)
= 8 (3.14)(22)(0.8) = 80 gms = 0.08 kg
Without including the mass of the striker the min speed
is: 2.86 m/s i.e. 10.3 kph
Including the mass of the striker, the min speed is: 3.5
m/s i.e. 12.6 kph
b)
Bamboo striker:
Similarly,
Ms = r (pr2t)
= 0.5 (3.14)(22)(2.2) = 13 gms = 0.013 kg
Without including the mass of the striker the min speed
is: 1.21 m/s i.e. 4.34 kph
Including the mass of the striker, the min speed is:
2.38 m/s i.e. 8.56 kph
Note: if the striker mass is included, the effective length L should
decrease, because of a shift in the position of the centre of mass – but
luckily it does not come into the formulas at all!
(ii)
Chimes were assumed to be immobile: how
good is that approximation, or is it even valid?
Effective area of chime: Ach = prL/2 =
(3.14)(0.85)(36)/2 = 48.0 cm2
Volume of chime: Vch = p [(r1
)2-(r2)2] L = (3.14)(0.852 – 0.702)
= 26 cc
Mass of chime mch = r Vch
= (8)(26) =210 gms = 0.21 kg
Force of gravity on chime: F = mg =
(9.8)(0.21) = 2.06 Nt
Force due to wind at 10 kph (2.78
m/s): F = Ach ( rair v2)/2 = (0.0048)(1.2)(2.78)2/2 =
0.022 Nt
F/mg = 0.022/2.06 = 0.0108
mg/F = 92.6
h = 2L/(1 + 92.62) =
2(0.75)/8573 = 1.75x10-4 m = 0.175 mm.
Tan(q/2) =
F/mg = 0.0108 q = 1.24°.
For a chime, the centre of mass is at L/2, so this
height and angle should be half (0.088 mm and 0.62°).
(iii)
Angular issues:
Another issue which was not mentioned earlier is the
angular factor, taking into account the number of chimes. In the metal chimes
that I have, there are 5 chimes, so that a top view would show a 72° angular separation between adjacent chimes when they
are at rest. The bamboo chimes have 6 chimes, so the angular separation between
adjacent chimes is 60° (as shown in the above figure). In the preceding derivation, the striker is assumed to
move away from the centre along a radius, and it is also assumed that the chime
is also located on the same radius – a case of motion along a line. However,
the striker may also move in a direction that takes it along a radius that is
midway between two adjacent chimes – or, indeed, at an arbitrary angle. The
displacement xm was earlier calculated using the linear case. The most
general case, if the striker moves at an arbitrary angle, is more difficult –
but the case in which it moves midway between two adjacent chimes is not so tough. In the above figure, the striker (of
radius s) moves a distance d, till it hits both chimes (of radius c) simultaneously. In the figure above, there are 6 chimes, so the positions of the chimes are separated by 60°.
In the case of linear motion (discussed previously), the condition for the striker to touch
the chime is:
d1 + s + c = r
where r is the radius of the circle on which the centres
of the 6 chimes are located.
Now consider the case in which the striker moves midway
between adjacent chimes in the figure above. Using the cosine law, the condition for the striker to
touch both chimes at once is:
(s+c)2 = (d2)2 + r2 – 2d2r cos(q)
where 2q = 60°.
One can plug in the numbers for s, c and r and solve the
quadratic to obtain the displacement d. This can
displacement then be put into the previously calculated xm and find
the wind-speed that will cause contact between the striker and the chime.
Solving the quadratic:
d2 = - rcos(q) + ((rcos(q))2 +
(s+c)2)1/2
Is d1 or d2 greater? That depends on the specific values of r,s,c and q - it
could go either way.
If the displacement required is greater in the midway
case (d2) than it is in the linear case (d1), then the wind-speed is also higher in the midway case (and vice
versa). For the case of 6 chimes, the plot of v(q) should
exhibit six-fold symmetry; and for 5 chimes, it will exhibit five-fold
symmetry.
(iv)
Miscellaneous:
Here it is assumed that the material of the chimes, the
striker, and the wind sail (bob) are all the same. To alleviate this problem,
the density of the bob & striker should be lower than that of the chimes.
Apparently, the material of the striker is different, but the chimes and the
bob seem to be of the same material in the metal chimes.
The Bali bamboo chimes have another odd problem: the
chimes are not kept hanging vertically down: they are at about 10-15° away
from the vertical, pointing radially outward from the bob/striker at the
centre.
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