Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2019

An inertial frame SS and another reference frame SS^{\prime} have a common origin OO, and SS^{\prime} rotates with angular velocity vector ω(t)\omega(t) with respect to SS. Derive the results (a) and (b) below, where dot denotes a derivative with respect to time tt :

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

(a˙)S=(a˙)S+ω×a.(\dot{\mathbf{a}})_{S}=(\dot{\mathbf{a}})_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{a} .

(b) The accelerations in SS and SS^{\prime} are related by

(r¨)S=(r¨)S+2ω×(r˙)S+(ω˙)S×r+ω×(ω×r),(\ddot{\mathbf{r}})_{S}=(\ddot{\mathbf{r}})_{S^{\prime}}+2 \boldsymbol{\omega} \times(\dot{\mathbf{r}})_{S^{\prime}}+(\dot{\boldsymbol{\omega}})_{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.

Just after passing the South Pole, a ski-doo of mass mm is travelling on a constant longitude with speed vv. Find the magnitude and direction of the sideways component of apparent force experienced by the ski-doo. [The sideways component is locally along the surface of the Earth and perpendicular to the motion of the ski-doo.]

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