Problem 3. (a) Show (as I will have done in class) that for two particles with only a force between them (and the force of 2 on 1 we call f) then the motion is described by an "internal" Newton's 2nd law, Vem = const, f= µÝrel where vrel = V1 – v2 = r1 - r2, and mįvi + m2V2 m1 + m2 mim2 cm mi + m2 (Since Taylor calls ri - r2 simply r in Chapter 8, he would probably also call Vrel simply v.) (b) Write an internal Lagrangian that yields this equation of motion.
Problem 3. (a) Show (as I will have done in class) that for two particles with only a force between them (and the force of 2 on 1 we call f) then the motion is described by an "internal" Newton's 2nd law, Vem = const, f= µÝrel where vrel = V1 – v2 = r1 - r2, and mįvi + m2V2 m1 + m2 mim2 cm mi + m2 (Since Taylor calls ri - r2 simply r in Chapter 8, he would probably also call Vrel simply v.) (b) Write an internal Lagrangian that yields this equation of motion.
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![**Problem 3.**
(a) Show (as I will have done in class) that for two particles with only a force between them (and the force of 2 on 1 we call \( \mathbf{f} \)) then the motion is described by an "internal" Newton’s 2nd law,
\[
\mathbf{v}_{cm} = \text{const.}, \quad \mathbf{f} = \mu \dot{\mathbf{v}}_{rel}
\]
where \( \mathbf{v}_{rel} = \mathbf{v}_1 - \mathbf{v}_2 = \dot{\mathbf{r}}_1 - \dot{\mathbf{r}}_2 \), and
\[
\mathbf{v}_{cm} = \frac{m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2}{m_1 + m_2}, \quad \mu = \frac{m_1 m_2}{m_1 + m_2}
\]
(Since Taylor calls \( \mathbf{r}_1 - \mathbf{r}_2 \) simply \( \mathbf{r} \) in Chapter 8, he would probably also call \( \mathbf{v}_{rel} \) simply \( \mathbf{v} \).)
(b) Write an internal Lagrangian that yields this equation of motion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2F4feb5faa-9771-4a53-bc84-cac226a8b9d4%2Foq4j18k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3.**
(a) Show (as I will have done in class) that for two particles with only a force between them (and the force of 2 on 1 we call \( \mathbf{f} \)) then the motion is described by an "internal" Newton’s 2nd law,
\[
\mathbf{v}_{cm} = \text{const.}, \quad \mathbf{f} = \mu \dot{\mathbf{v}}_{rel}
\]
where \( \mathbf{v}_{rel} = \mathbf{v}_1 - \mathbf{v}_2 = \dot{\mathbf{r}}_1 - \dot{\mathbf{r}}_2 \), and
\[
\mathbf{v}_{cm} = \frac{m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2}{m_1 + m_2}, \quad \mu = \frac{m_1 m_2}{m_1 + m_2}
\]
(Since Taylor calls \( \mathbf{r}_1 - \mathbf{r}_2 \) simply \( \mathbf{r} \) in Chapter 8, he would probably also call \( \mathbf{v}_{rel} \) simply \( \mathbf{v} \).)
(b) Write an internal Lagrangian that yields this equation of motion.
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