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A frame of 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 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 \(\vec{F}_{\text {in }}\) experienced by the particle in an inertial frame of reference is

\(\vec{F}_{\mathrm{rot}}=\vec{F}_{\mathrm{in}}+2 m\left(\vec{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega}\),

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 \(R\) 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 \((\vec{\omega}=\omega \hat{k}) .\) A small block of mass \(m\) is gently placed in the slot at \(\vec{r}=(R / 2) \hat{i}\) at \(t=0\) and is constrained to move only along the slot.

Question :
The net reaction of the disc on the block is

1. A $-m{\omega }^{2}R\cos \omega t\widehat{j}-mg\widehat{k}$
2. B $m{\omega }^{2}R\sin \omega t \widehat{j}-mg\widehat{k}$
3. C $\frac{1}{2}m{\omega }^{2}R\left(e^{\omega t}-e^{-\omega t}\right) \widehat{j}+mg\widehat{k}$
4. D $\frac{1}{2}m{\omega }^{2}R \left(e^{2\omega t}-e^{-2\omega t}\right)\widehat{j}+mg\widehat{k}$