P15_007 - 7. (a) The pressure difference results in forces...

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Unformatted text preview: 7. (a) The pressure difference results in forces applied as shown in the figure. We consider a team of horses pulling to the right. To pull the sphere apart, the team must exert a force at least as great as the horizontal component of the total force determined by “summing” (actually, integrating) these force vectors. .. . . .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. ........... .......... .. . ... .... ..... ..... ..... ..... .... ...... .... ..... . .... .... ....... .. .. ..... .... .... . .. .... . . ..... .... . ... . . .. .... .. . .. . ....... . .. ........ . .. ..... . . . ....... ...... .. .. . .. .. . . . . . . . .. . . .. . . . . . . . . . . . .. . .. . . . .. . . .. . . . .... ... ... ... . ... ... ...................................... .... .... .... . .... .... ...................................... . . .. . .. .. .. . . . . . . . . . . . . . .. ... ... ........ ........ .. . ..... .. . ....... .... .. .... ... .... .. .. .... .... ... ... .... .... ... ... . .... . . .... ....... ... . .... ......... .. . .. ... . .... ... ....... .. ....... .. ..... ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . We consider a force vector at angle θ. Its leftward component is ∆p cos θdA, where dA is the area element for where the force is applied. We make use of the symmetry of the problem and let dA be that of a ring of constant θ on the surface. The radius of the ring is r = R sin θ, where R is the radius of the sphere. If the angular width of the ring is dθ, in radians, then its width is R dθ and its area is dA = 2πR2 sin θ dθ. Thus the net horizontal component of the force of the air is given by r • θ R π /2 Fh = 2πR2 ∆p sin θ cos θ dθ 0 = πR2 ∆p sin2 θ π /2 = πR2 ∆p . 0 (b) We use 1 atm = 1.01 × 105 Pa to show that ∆p = 0.90 atm = 9.09 × 104 Pa. The sphere radius is R = 0.30 m, so Fh = π (0.30 m)2 (9.09 × 104 Pa) = 2.6 × 104 N. (c) One team of horses could be used if one half of the sphere is attached to a sturdy wall. The force of the wall on the sphere would balance the force of the horses. ...
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