Paper 4, Section II, 12C12 \mathrm{C}

Dynamics | Part IA, 2007

The ii th particle of a system of NN particles has mass mim_{i} and, at time tt, position vector ri\mathbf{r}_{i} with respect to an origin OO. It experiences an external force Fie\mathbf{F}_{i}^{e}, and also an internal force Fij\mathbf{F}_{i j} due to the jj th particle (for each j=1,,N,jij=1, \ldots, N, j \neq i ), where Fij\mathbf{F}_{i j} is parallel to rirj\mathbf{r}_{i}-\mathbf{r}_{j} and Newton's third law applies.

(i) Show that the position of the centre of mass, X\mathbf{X}, satisfies

Md2Xdt2=FeM \frac{d^{2} \mathbf{X}}{d t^{2}}=\mathbf{F}^{e}

where MM is the total mass of the system and Fe\mathbf{F}^{e} is the sum of the external forces.

(ii) Show that the total angular momentum of the system about the origin, L\mathbf{L}, satisfies

dLdt=N\frac{d \mathbf{L}}{d t}=\mathbf{N}

where N\mathbf{N} is the total moment about the origin of the external forces.

(iii) Show that L\mathbf{L} can be expressed in the form

L=MX×V+imiri×vi\mathbf{L}=M \mathbf{X} \times \mathbf{V}+\sum_{i} m_{i} \mathbf{r}_{i}^{\prime} \times \mathbf{v}_{i}^{\prime}

where V\mathbf{V} is the velocity of the centre of mass, ri\mathbf{r}_{i}^{\prime} is the position vector of the ii th particle relative to the centre of mass, and vi\mathbf{v}_{i}^{\prime} is the velocity of the ii th particle relative to the centre of mass.

(iv) In the case N=2N=2 when the internal forces are derived from a potential U(r)U(|\mathbf{r}|), where r=r1r2\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}, and there are no external forces, show that

dTdt+dUdt=0\frac{d T}{d t}+\frac{d U}{d t}=0

where TT is the total kinetic energy of the system.

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