Dynamics | Part IA, 2008

An inertial reference frame SS and another reference frame SS^{\prime} have a common origin OO, and SS^{\prime} rotates with angular velocity ω(t)\boldsymbol{\omega}(t) with respect to SS. Show the following:

(i) the rates of change of an arbitrary vector a (t)(t) in frames SS and SS^{\prime} are related by

(dadt)S=(dadt)S+ω×a\left(\frac{d \mathbf{a}}{d t}\right)_{S}=\left(\frac{d \mathbf{a}}{d t}\right)_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{a}

(ii) the accelerations in SS and SS^{\prime} are related by

(d2rdt2)S=(d2rdt2)S+2ω×(drdt)S+(dωdt)S×r+ω×(ω×r)\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}=\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}+2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}+\left(\frac{d \boldsymbol{\omega}}{d t}\right)_{S^{\prime}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})

where r(t)\mathbf{r}(t) is the position vector relative to OO.

A train of mass mm at latitude λ\lambda in the Northern hemisphere travels North with constant speed VV along a track which runs North-South. Find the magnitude and direction of the sideways force exerted on the train by the track.

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