Paper 4, Section II, C

Dynamics | Part IA, 2007

A particle of mass mm experiences, at the point with position vector r\mathbf{r}, a force F\mathbf{F} given by

F=krer˙×B,\mathbf{F}=-k \mathbf{r}-e \dot{\mathbf{r}} \times \mathbf{B},

where kk and ee are positive constants and B\mathbf{B} is a constant, uniform, vector field.

(i) Show that mr˙r˙+krrm \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}+k \mathbf{r} \cdot \mathbf{r} is constant. Give a physical interpretation of each term and a physical explanation of the fact that B\mathbf{B} does not arise in this expression.

(ii) Show that m(r˙×r)B+12e(r×B)(r×B)m(\dot{\mathbf{r}} \times \mathbf{r}) \cdot \mathbf{B}+\frac{1}{2} e(\mathbf{r} \times \mathbf{B}) \cdot(\mathbf{r} \times \mathbf{B}) is constant.

(iii) Given that the particle was initially at rest at r0\mathbf{r}_{0}, derive an expression for rB\mathbf{r} \cdot \mathbf{B} at time tt.

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