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homework1sols - Chapter 13 Problems 6. Using F = GmM/r2, we...

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Chapter 13 Problems 6. Using F = GmM/r 2 , we find that the topmost mass pulls upward on the one at the origin with 1.9 × 10 8 N, and the rightmost mass pulls rightward on the one at the origin with 1.0 × 10 8 N. Thus, the ( x, y ) components of the net force, which can be converted to polar components (here we use magnitude-angle notation), are () ( ) 88 8 net = 1.04 10 ,1.85 10 2.13 10 60.6 . F −− ×× × ° G (a) The magnitude of the force is 2.13 × 10 8 N. (b) The direction of the force relative to the + x axis is 60.6 ° . 11. (a) The distance between any of the spheres at the corners and the sphere at the center is /2cos30 / 3 r = AA where A is the length of one side of the equilateral triangle. The net (downward) contribution caused by the two bottom-most spheres (each of mass m ) to the total force on m 4 has magnitude 44 22 2= 2 s i n 3 0 = 3 . y Gm m Gm m F r ⎛⎞ ° ⎜⎟ ⎝⎠ A This must equal the magnitude of the pull from M , so 2 2 3 /3 Gm m Gm m = A A which readily yields m = M . (b) Since m 4 cancels in that last step, then the amount of mass in the center sphere is not relevant to the problem. The net force is still zero. 85. We use m 1 for the 20 kg of the sphere at ( x 1 , y 1 ) = (0.5, 1.0) (SI units understood), m 2 for the 40 kg of the sphere at ( x 2 , y 2 ) = ( 1.0, 1.0), and m 3 for the 60 kg of the sphere at ( x 3 , y 3 ) = (0, 0.5). The mass of the 20 kg object at the origin is simply denoted m . We note that 12 1.25, 2 rr == , and r 3 = 0.5 (again, with SI units understood). The force n F G that the n th sphere exerts on m has magnitude 2 / nn Gm m r and is directed from the origin towards m n , so that it is conveniently written as
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() 23 ˆˆ ˆ ˆ =i + j = i + j . nn n n n nnn n Gm m x y Gm m Fx y rrr r ⎛⎞ ⎜⎟ ⎝⎠ G Consequently, the vector addition to obtain the net force on m becomes 33 3 97 net =1 1 1 ˆ ˆ = i j 9.3 10 i 3.2 10 j n n mx my FF G m rr −− == =+ = × × ∑∑ GG in SI units. Therefore, we find the net force magnitude is 7 net 3.2 10 N F G .
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homework1sols - Chapter 13 Problems 6. Using F = GmM/r2, we...

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