11-7-08LectPHY2048

# 11-7-08LectPHY2048 - One more example in statics The two...

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One more example in statics The two spheres, each of mass m, are at rest in the box as shown. With friction negligible what are the forces on the spheres from the bottom left and right walls and on each other?

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mg L F B F 2,1y F 2,1x F 1,2y F 1,2x F R F 1 2 Draw all the forces acting on the two spheres and resolve the forces from the spheres on each other into their x and y components. Then for sphere 2 along y: 2,net,y F0 = Fm g 0 −= 1,2 F 2,1 F g = But also o 1,2 FF c o s ( 4 5 ) = So, 1,2 o mg F2 m g cos(45 ) == (the force of the spheres on each other)
mg L F B F 2,1y F 2,1x F 1,2y F 1,2x F R F 1 2 1,2 F 2,1 F For sphere 2 along x, 1,2x R FF 0 += o R1 , 2 x1 , 2 F s i n ( 4 5 ) =− o R o mg Fs i n ( 4 5 ) cos(45 ) But we found above that, 1,2 o mg F cos(45 ) = So, o R Fm g t a n ( 4 5 ) m g 1

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mg L F B F 2,1y F 2,1x F 1,2y F 1,2x F R F 1 2 1,2 F 2,1 F For sphere 1 along y, B2 , 1 y Fm g F 0 −+ = o , 1 g Fc o s ( 4 5 ) =− But we must have (see figure), 2,1 1,2 o mg FF cos(45 ) o B o mg g cos(45 ) cos(45 ) So, B g m g 2 m g =+= Which we might have realized immediately since there is no friction ( F B is the only vertical force supporting the weight of both spheres).
mg L F B F 2,1y F 2,1x F 1,2y F 1,2x F R F 1 2 1,2 F 2,1 F For sphere 1 along x, L2 , 1 x FF 0 += , 1 x =− o , 1 F F sin(45 ) o L o mg Fs i n ( 4 5 ) cos(45 ) o L Fm g t a n ( 4 5 ) m g == Which we could have gotten immediately from F R since the forces by the spheres on each other cancel ( F L and F R are the only external forces acting on the two spheres taken as a unit).

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Gravitation By the time of Newton (1643 – 1727) it was largely recognized that the earth was round and the planets orbited about our sun . It was also obvious that the earth exerted some force on objects near its surface . Dropped objects (e.g. apples) fell to the ground. What was completely non-obvious was the idea that the force keeping the planets in their orbit and the force causing objects on earth to fall had exactly the same underlying mechanism : gravity . So how did Newton arrive at this idea that matter attracts all other matter. The recognition that all massive matter attracts all other massive matter with a force that can be quantitatively expressed in just a few symbols stands as one of the great intellectual achievements in human history.
Having formulated the three laws of mechanics Newton recognized, while contemplating the nearly circular orbit of the moon about the earth, that the circular orbit implied a centripetal acceleration inward toward earth’s center. Since F = ma , that acceleration implied a force toward the center, i.e. toward the earth . He next had the brilliance to wonder (and having invented calculus, the mathematical prowess to calculate) whether or not this force could be connected to the force that caused objects near earth’s surface to fall. His conclusion was that any two material objects exert an attractive force on each other, in direct proportion to their masses and in inverse proportion to the square of the distance between them .

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11-7-08LectPHY2048 - One more example in statics The two...

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