Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2010

Consider a set of particles with position vectors ri(t)\mathbf{r}_{i}(t) and masses mim_{i}, where i=1,2,,Ni=1,2, \ldots, N. Particle ii experiences an external force Fi\mathbf{F}_{i} and an internal force Fij\mathbf{F}_{i j} from particle jj, for each jij \neq i. Stating clearly any assumptions you need, show that

dPdt=F and dLdt=G\frac{d \mathbf{P}}{d t}=\mathbf{F} \quad \text { and } \quad \frac{d \mathbf{L}}{d t}=\mathbf{G}

where P\mathbf{P} is the total momentum, F\mathbf{F} is the total external force, L\mathbf{L} is the total angular momentum about a fixed point a\mathbf{a}, and G\mathbf{G} is the total external torque about a\mathbf{a}.

Does the result dLdt=G\frac{d \mathbf{L}}{d t}=\mathbf{G} still hold if the fixed point a\mathbf{a} is replaced by the centre of mass of the system? Justify your answer.

Suppose now that the external force on particle ii is kdridt-k \frac{d \mathbf{r}_{i}}{d t} and that all the particles have the same mass mm. Show that

L(t)=L(0)ekt/m\mathbf{L}(t)=\mathbf{L}(0) e^{-k t / m}

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