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 non=inertial frame of reference. The relationship between the force `vecF_(rot)` experienced by a particle of mass m moving on the rotating disc and the force `vecF_(in)` experienced by the particle in an inertial frame of reference is
`vecF_(rot)=vecF_(i n)+2m(vecv_(rot)xxvec omega)+m(vec omegaxx vec r)xxvec omega`.
where `vecv_(rot)` is the velocity of the particle in the rotating frame of reference and `vecr` is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter fo 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 `(vecomega=omegahatk)`. A small block of mass m is gently placed in the slot at `vecr(R//2)hati` at `t=0` and is constrained to move only along the slot.

The net reaction of the disc on the block is

Options

A. `-m omega^(2)R cos omega t hatj- mghatk`
B. `- omega^(2)R(e^(2omegat)-e^(-2 omegat)) hatj+mg hatk`
C. `momega^(2)R sin omega t hatj- mg hatk`
D. `(1)/(2) m omega^(2)R (e^( omegat- e^(- omegat)) hatj+ mghatk)`

Answer

D

Explanation

Net normal reaction on the disc can be writtenas follows :
`vecN= N_(1) hatj+N_(2) hatk`
`vecN= 2m omega (dr)/(dt) hatj+mghatk`
`rArr vecN=2 m omega(d)/(dt) [(R)/(4) (e^(omega t)+e^(- omegat))] hatj+ mghatk`
`rArr vec N=(1)/(2) m R omega^(2) ((e^(omega t)+e^(- omegat)) hatj+ mg hatk`
Hence option (d) is correct.