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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 \( \vec{F}_{\text {rot }} \) experienced by a particle of mass \( m \) moving on the rotating disc and the force \( \overrightarrow{\mathrm{F}}_{\text {in }} \) 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 \( \vec{v}_{\text {tot }} \) 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 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 \( x \)-axis along the slot, the \( y \)-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 \( \vec{r}=(R / 2) \hat{i} \)
at \( \mathrm{t}=0 \) and is constrained to move only along the Slot.
The net reaction of the disc on the block is
(A) \( \frac{1}{2} m \omega^{2} R\left(e^{\omega m}-e^{-\omega \omega}\right) \hat{j}+m g \hat{k} \)
(B) \( -m \omega^{2} R \cos \omega \hat{j}-m g \hat{k} \)
(C) \( \frac{1}{2} m \omega^{2} R\left(e^{2 \omega t}-e^{-2 e x}\right) \hat{j}+m g \hat{k} \)
(D) \( m \omega^{2} R \sin \omega \hat{j}-m g \hat{k} \)
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