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A frame of reference that is accelerated with respect to an inertial frame of reference is called a noninertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity \( \omega \) is an example of a non-inertial frame of reference.
The relationship between the force \( \vec{F}_{\mathrm{rot}} \) experienced by a particle of mass \( m \) moving on the rotating disc and the force \( \overrightarrow{\mathrm{F}}_{\mathrm{in}} \) experienced by the particle in an inertial frame of reference is
\[
\overrightarrow{\mathrm{F}}_{\text {rot }}=\overrightarrow{\mathrm{F}}_{\text {in }}+2 \mathrm{~m}\left(\overrightarrow{\mathrm{v}}_{\text {rot }} \times \vec{\omega}\right)+\mathrm{m}(\vec{\omega} \times \overrightarrow{\mathrm{r}}) \times \vec{\omega},
\]
where \( \vec{v}_{\text {rot }} \) is the velocity of the particle in the rotating frame of reference and \( \vec{r} \) is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter of a disc of radius \( \mathrm{R} \) rotating counter-clockwise with a constant angular speed \( \omega \) about its vertical axis through its center. We assign a coordinate system with - the origin at the center of the disc, the \( \mathrm{x} \)-axis along the slot, the \( \mathrm{y} \)-axis perpendicular to the slot and the
\( z \)-axis along the rotation axis \( (\vec{\omega}=\omega \hat{\mathrm{k}}) \). Asmall block of mass m is gently placed in the slot at \( \overrightarrow{\mathrm{r}}=(\mathrm{R} / 2) \hat{\mathrm{i}} \)
at \( \mathrm{t}=0 \) and is constrained to move only along the Slot.
The distance r of the block at time \( t \) is
(A) \( \frac{\mathrm{R}}{2} \cos \omega \mathrm{t} \)
(B) \( \frac{\mathrm{R}}{4}\left(\mathrm{e}^{200 t}+\mathrm{e}^{-2 e t}\right) \)
(C) \( \frac{\mathrm{R}}{4}\left(\mathrm{e}^{\mathrm{mt}}+\mathrm{e}^{-\mathrm{mt}}\right) \)
(D) \( \frac{\mathrm{R}}{2} \cos 2 \omega t \)
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