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s11f1017

s11f1017 - 1 2 μ~v 2 4 Angular momentum m 1 ~ r 1 × ~v 1...

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Solution 17 set 4. 1. Yes, the Lagrangian does not depend on the time. 2. Solving for ~ r 1 : M ~ R = m 1 ~ r 1 + m 2 ~ r 2 , M = m 1 + m 2 , ~ r = ~ r 1 - ~ r 2 , m 1 ~ r 1 = M ~ R - m 2 ~ r 2 = M ~ R + m 2 ~ r - m 2 ~ r 1 , ~ r 1 = ~ R + m 2 M ~ r , m 2 ~ r 2 = M ~ R - m 1 ~ r 1 = M ~ R - m 1 ~ r - m 1 ~ r 2 , ~ r 2 = ~ R - m 1 M ~ r . 3. In the CM frame ~ R = 0 and using μ = m 1 m 2 / ( m 1 + m 2 ) we can write ~ r 1 = μ m 1 ~ r , ~ r 2 = - μ m 2 ~ r . Therefore, T cm = 1 2 ± μ 2 m 1 + μ 2 m 2 ! ~v 2 = 1 2 μ 2 ( m 1 + m 2 ) m 1 m 2 ~v 2 =
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Unformatted text preview: 1 2 μ~v 2 4. Angular momentum: m 1 ~ r 1 × ~v 1 + m 2 ~ r 2 × ~v 2 = m 1 μ m 1 ~ r × ~v 1-m 2 μ m 2 ~ r × ~v 2 = μ~ r × ( ~v 1-~v 2 ) = μ~ r × ~v ....
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