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 "sin" omega t hatj - mg hat k `
B. `(1)/(2) m omega^(2) R(e^(omega t ) - e^(- omega t )) hat j + mg hat k`
C. `(1)/(2) m omega^(2) R (e^(2 omega t) - e^(-2 omega t)) hat j + mg hat k `
D. `- m omega^(2) R "cos" omega t hatj - mg hatk `

Answer

B

Explanation

`vecv_("rot") = (d vecr)/(dt) = (R)/(4) omega [e^(omega t) - e^(- omega t )]hati`
`therefore " " 2m(vecv_("rot") xx vecomega)`
`vecF_("rot") = 2m( R omega^(2))/(4)[e^(omega t) - e^(- omega t )](-hatj)`
Reaction force = `(1)/(2) momega^(2) R [e^(omega t) - e^(- omega t )]hatj + mg hatk`