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HW1_prob2_GPS02_G05

HW1_prob2_GPS02_G05 - result M 2 R 2 = 1 2 X i X j m i m j...

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Problem 1.5 (3 rd ed. 1.2) Start with the de f nition of the center of mass vector R , M R = N X i =1 m i r i , (1) where m i and r i are, respectively, the mass and position vector of particle i and M is the total mass, M = P i m i . Next, take the dot product of Eq.(1) with itself to get M 2 R 2 = X i X j m i m j r i · r j . (2) Now take the dot product of r ij with itself, where r ij = r i r j . This results in r ij · r ij = r 2 ij = r 2 i + r 2 j 2 r i · r j . (3) Next, replace r i · r j in Eq.(2) with the result in Eq.(3) to obtain the intermediate
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Unformatted text preview: result, M 2 R 2 = 1 2 X i X j m i m j r 2 i + 1 2 X i X j m i m j r 2 j − 1 2 X i X j m i m j r 2 ij . (4) The f rst two pairs of sums in Eq.(4) are identical because the indices i and j run over all values from 1 to N . Thus, this equation simplies immediately to the desired result, M 2 R 2 = M X j m j r 2 j − 1 2 X i X j m i m j r 2 ij . (5) 1...
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