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Passage 1
A frame of the reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular dise rotating about a fixed axis with a constant angular velocity \( \omega \) is
\( P \)
W an example of a non-inertial frame of reference. The relationship between the force \( \mathbf{F}_{\text {rot }} \) experienced by a particle of mass \( m \) moving on the rotating disc and the force \( F_{\text {in }} \) experienced by the particle in an inertial frame of reference is, \( \mathbf{F}_{\text {rot }}=\mathbf{F}_{\text {in }}+2 m\left(\mathbf{v}_{\text {rot }} \times \vec{\omega}\right)+m(\vec{\omega} \times \mathbf{r}) \times \vec{\omega} \), where, \( \mathbf{v}_{\mathrm{rot}} \) is the velocity of the particle in the rotating frame of reference and \( \mathbf{r} \) 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 centre. We assign a coordinate system with the origin at the centre of the disc, the \( X \)-axis along the slot, the \( Y \)-axis perpendicular to the slot and the \( z \)-axis along th rotation axis \( (\omega=\omega \mathbf{k}) \). A small block of mass \( m \) is gently placed in the slot at \( \mathbf{r}=(R / 2) \hat{\mathbf{i}} \) at \( t=0 \) and is constrained to move only along the slot. (2016 Adv.)
The distance \( r \) of the block at time \( t \) is
(a) \( \frac{R}{2} \cos 2 \omega t \)
(b) \( \frac{R}{2} \cos \omega t \)
(c) \( \frac{R}{4}\left(e^{\omega t}+e^{-\omega t}\right) \)
(d) \( \frac{R}{4}\left(e^{2 \omega t}+e^{-2 \omega t}\right) \)
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