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265_Physics ProblemsTechnical Physics

# 265_Physics ProblemsTechnical Physics - Chapter 9*P9.42(a...

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Chapter 9 267 *P9.42 (a) Represent the height of a particle of mass dm within the object as y . Its contribution to the gravitational energy of the object-Earth system is dm gy a f . The total gravitational energy is U gydm g ydm g = = z z all mass . For the center of mass we have y M ydm CM = z 1 , so U gMy g = CM . (b) The volume of the ramp is 1 2 3 6 15 7 64 8 1 83 10 3 . . . . m m m m 3 a fa fa f = × . Its mass is ρ V = × = × 3 800 1 83 10 6 96 10 3 6 kg m m kg 3 3 e je j . . . Its center of mass is above its base by one- third of its height, y CM m m = = 1 3 15 7 5 23 . . . Then U Mgy g = = × = × CM 2 kg m s m J 6 96 10 9 8 5 23 3 57 10 6 8 . . . . e j . P9.43 (a) M dx x dx = = + z z λ 0 0 300 0 0 300 50 0 20 0 . . . . m 2 m g m g m M x x = + = 50 0 10 0 15 9 2 0 0 300 . . . . g m g m g 2 m (b) x xdm M M xdx M x x dx CM all mass m 2 m g m g m = = = + z z z 1 1 50 0 20 0 0 0 300 2 0 0 300 λ . . . . x x x CM 2 m g g m g m m = + L N M M O Q P P = 1 15 9 25 0 20 3 0 153 2 3 0 0 300 . . . . *P9.44 Take the origin at the center of curvature. We have L r = 1 4 2 π , r L = 2 π . An incremental bit of the rod at angle
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