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Hexagonal sphere packing Imagine three unit spheres
Chapter 11, Problem 65E(choose chapter or problem)
Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0), P\((\sqrt{3},-1,0)\), and \(Q(\sqrt{3}, 1,0)\). Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure).
a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.)
b. Let \(\mathbf{r}_{i j}\) be the vector from the center of sphere i to the center of sphere j. Find \(\mathbf{r}_{O P}, \mathbf{r}_{O Q}, \mathbf{r}_{P Q}, \mathbf{r}_{O R}\), and \(\mathbf{r}_{P R}\).
Questions & Answers
QUESTION:
Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0), P\((\sqrt{3},-1,0)\), and \(Q(\sqrt{3}, 1,0)\). Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure).
a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.)
b. Let \(\mathbf{r}_{i j}\) be the vector from the center of sphere i to the center of sphere j. Find \(\mathbf{r}_{O P}, \mathbf{r}_{O Q}, \mathbf{r}_{P Q}, \mathbf{r}_{O R}\), and \(\mathbf{r}_{P R}\).
ANSWER:Solution 65E
Step 1 of 2 :
The given figure is ;
a) The center of the circles are O (0 ,0,0) , P ( , -1 , 0) , Q (
, 1 , 0 ) and center R of another unit sphere symmetrically is to find .
Given that , the distance between the centers of any two spheres is 2.
Clearly the points O (0 ,0,0) , P ( , -1 , 0) and Q (
, 1 , 0 ) form an equilateral triangle.
Therefore , the point R should lie on the straight line parallel to z - axis and passing through the centroid of the .
Therefore , centroid is = ( ,
,
)
= ( , 0 ,0).
Therefore , R = ( , 0 ,t) , where t > 0
Given OR = 2, so by the distance formula ; = 2
Squaring on both sides we get , +
= 4
= 4 -
=
=
Hence , t =
Therefore , R = ( , 0 ,
)
Note ; The distance between two points is square root of sum of square of the difference of corresponding coordinates. |