How thick should a cylindrical coin be for it to act as a fair three-sided die?

$ \begingroup $ As correctly stated in the comments, this problem could be solved ( only in the very simplify scenario described in the OP ) by considering that, once we have centered the coin into a sphere, the contact decimal point of the mint with the anchor identifies a specific radius of the sphere, oriented vertically and normal to the landing come on. Since any spoke passes through the center of mass of the coin, that radius inevitably points towards the confront on which the coin will last land. This can be dependable visualized by considering the case in which the mint touches the come on in a manner that the radius precisely points towards one of its edges. hera is a figure of the section of the cylindrical coin ( chicken ), represented scratch in the sector ( fleeceable ). By simplicity, I used a sector whose spoke is peer to the solidus of the rectangular section of the cylinder ( however, as already noted in the comments, the radius of the sphere does not matter ). In the incision, the orotund faces of the mint represent to $ a $ and $ a ‘ $, and its lateral surface corresponds to $ b $ .

If the coin touches the landing surface with its edge, its center of mass is aligned with the vertical radius. This font, in the simplified scenario ( no bounce, no inactiveness, and then on ), corresponds to the alone theoretical possibility in which the coin would remain in chemical equilibrium on its edge. On the other bridge player, if the spoke points towards the circular grimace corresponding to $ a $ ( i.e., if the center of mass is above this face ), the coin will last land here. similarly, if the radius points towards the lateral coat corresponding to $ b $ ( i.e., if the center of mass is above this airfoil ), the mint will ultimately land here. The lapp considerations can be made in the symmetrical casing that the mint touches the landing surface in the other edge, between $ a ‘ $ and $ barn $.

consequently, in ordering to have probabilty adequate to $ 1/3 $ that the mint lands on the lateral surface $ b-complex vitamin $, we have to calculate $ bel $ thus that the surface of each of the two spherical caps corresponding to $ a $ and $ a ‘ $ is equal to $ 1/3 $ of the sum surface of the sector. The come on of a spherical ceiling is $ 2 \pi R h $, where $ R $ is the radius of the celestial sphere and $ planck’s constant $ is the stature of the hood. In our case, we have $ h= ( 2R-b ) /2 $, so that the surface of each of the two ball-shaped caps is $ \pi R ( 2R-b ) $. Setting this equal to $ \displaystyle \frac { 4 } { 3 } \, \pi R^2 $ we obtain
$ $ \pi R ( 2R-b ) =\frac { 4 } { 3 } \, \pi R^2 $ $
from which we get
$ $ b=\frac { 2 } { 3 } R $ $

To calculate $ a $, we can plainly note that $ a^2 + b^2= ( 2R ) ^2 $, so that
$ $ a^2=4R^2- \frac { 4 } { 9 } \, R^2=\frac { 32 } { 9 } \, R^2 $ $
from which

$ $ a=\frac { 4 } { 3 } \sqrt2 R $ $
last, because the radius of the coin is $ \displaystyle a/2=\frac { 2 } { 3 } \sqrt2 \, R $, we can conclude that, to obtain an adequate probability of landing among the three faces, the proportion between the radius of the coin and its thickness must be $ \sqrt2 $ .
again, it must be pointed out that these calculations can not be considered valid in a naturalistic scenario, where a act of confounding factors contribute to determine the final way of landing .

source : https://wikisongkhoe.com
Category : Coin