Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2016

State what the vectors a,r,v\mathbf{a}, \mathbf{r}, \mathbf{v} and ω\boldsymbol{\omega} represent in the following equation:

a=g2ω×vω×(ω×r)\mathbf{a}=\mathbf{g}-2 \boldsymbol{\omega} \times \mathbf{v}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})

where g\mathbf{g} is the acceleration due to gravity.

Assume that the radius of the Earth is 6×106 m6 \times 10^{6} \mathrm{~m}, that g=10 ms2|\mathrm{g}|=10 \mathrm{~ms}^{-2}, and that there are 9×1049 \times 10^{4} seconds in a day. Use these data to determine roughly the order of magnitude of each term on the right hand side of ()(*) in the case of a particle dropped from a point at height 20 m20 \mathrm{~m} above the surface of the Earth.

Taking again g=10 ms2|\mathbf{g}|=10 \mathrm{~ms}^{-2}, find the time TT of the particle's fall in the absence of rotation.

Use a suitable approximation scheme to show that

RR013ω×gT312ω×(ω×R0)T2,\mathbf{R} \approx \mathbf{R}_{0}-\frac{1}{3} \boldsymbol{\omega} \times \mathbf{g} T^{3}-\frac{1}{2} \boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{R}_{0}\right) T^{2},

where R\mathbf{R} is the position vector of the point at which the particle lands, and R0\mathbf{R}_{0} is the position vector of the point at which the particle would have landed in the absence of rotation.

The particle is dropped at latitude 4545^{\circ}. Find expressions for the approximate northerly and easterly displacements of R\mathbf{R} from R0\mathbf{R}_{0} in terms of ω,g,R0\omega, g, R_{0} (the magnitudes of ω,g\boldsymbol{\omega}, \mathbf{g} and R0\mathbf{R}_{0}, respectively), and TT. You should ignore the curvature of the Earth's surface.

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