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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 disc 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 \( \mathrm{F}_{\text {rot }} \) 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, \( \mathbf{F}_{\mathrm{rot}}=\mathbf{F}_{\mathrm{in}}+2 m\left(\mathbf{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \mathbf{r}) \times \vec{\omega} \), where, \( \mathbf{v}_{\text {rot }} \) 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. \( \quad \) (2016 Adv.) The net reaction of the disc on the block is
(a) \( m \omega^{2} R \sin \omega \hat{\mathbf{j}}-m g \hat{\mathbf{k}} \)
(b) \( \frac{1}{2} m \omega^{2} R\left(e^{\omega}-e^{-\omega t}\right) \hat{\mathbf{j}}+m g \hat{\mathbf{k}} \)
(c) \( \frac{1}{2} m \omega^{2} R\left(e^{2 \omega x}-e^{-2 \omega x}\right) \hat{\mathbf{j}}+m g \hat{\mathbf{k}} \)
(d) \( -m \omega^{2} R \cos \omega \hat{\mathbf{j}}-m g \hat{\mathbf{k}} \)
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