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 .
Đây là website tự động và trong giai đoạn thử nghiệm tool tự động lấy bài viết, mọi thông tin đăng tải trên website này chúng tôi không chịu trách nhiệm dưới mọi hình thức, đây không phải là một website phát triển thông tin, nó được xây dựng lên với mục đích thử nghiệm các phương pháp tự động của chúng tôi mà thôi. Nếu có khiếu nại vui lòng gửi thông tin cho chúng tôi.