2/116. A force F is applied to the surface of the sphere as shown. The angles an

Question

2/116. A force F is applied to the surface of the sphere as shown. The angles and locate point P, and point M is the midpoint of ON. Express F in vector form, using the given x-,y-, and z-coordinates.

The book gives the answer for even number problems. The answer is:

F = F[(2sin-1)(icos + jsin) + k(2coq)/sgrt(5-4sin)]

I get the coordinates for Point P but I am not sure where to go from there.

2/116. A force F is applied to the surface of the sphere as shown. The angles theta and phi locate point P, and point M is the midpoint of ON. Express F in vector form, using the given x-,y-, and z-coordinates. The book gives the answer for even number problems. The answer is: F = F[(2sin phi -1)(icos theta + jsin theta ) + k(2coq phi )/sgrt(5-4sin phi )] I get the coordinates for Point P but I am not sure where to go from there.

Explanation / Answer

P(Rsincos, Rsinsin, Rcos)

M(Rcos/2, Rsin/2, 0)

vector MP = <Rsincos - Rcos/2, Rsinsin - Rsin/2, Rcos - 0>

= <(R/2)cos(2sin - 1), (R/2)sin(2sin - 1), Rcos>

magnitude of vector MP = |MP| = {[(R/2)cos(2sin - 1)]2 +[(R/2)sin(2sin - 1)]2 + (Rcos)2}

= (R/2) [(2sin - 1)2 + 4cos2] = (R/2) (5 - 4sin)

unit vector u (along MP, or along F) = <(R/2)cos(2sin - 1), (R/2)sin(2sin - 1), Rcos>/|MP|

= <cos(2sin - 1), sin(2sin - 1), 2cos>/(5 - 4sin)

so vector F = F*u = F[cos(2sin - 1) i + sin(2sin - 1) j + 2cos k]/(5 - 4sin)

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