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 distance r of the block at time is

Options

A. `(R)/(2) "cos" 2 omega t`
B. `(R)/(2) "cos" omega t `
C. `(R)/(4) (e^(omegat ) + e^(-omega t )) `
D. `(R)/(4) (e^(2omega t ) + e^(-2 omega t ))`

Answer

C

Explanation

Let `vecv` be the velocity of block at a distance r from the axis .
Now , `" " vecF_("in") = 0 , vecv_("rot") = v hati`
`vecr = rhati , vecomega = omega hatk`
Now , `(vecv_("rot") xx vecomega) = v omega (hati xx hatk) = - vomega hatj`
`(vecomega xx vecr) xx vecomega = omega^(2)rhati`
`therefore " " vecF_("rot") = -2mv omega hatj + m omega^(2) rhatj`
Now considering motion along x - axis
`m(dv)/(dt) = m omega^(2)r`
`mv(dv)/(dr) = m omega^(2)r`
`int_(0)^(v)int v dv = int_(r_(0))^(r) omega^(2) r dr`
`(v^(2))/(r) = (omega^(2) (r^(2) - r_(0)^(2)))/(2)`
`implies " " v^(2) = omega^(2) (r^(2) - r_(0)^(2))`
`implies " " v = omega sqrt(r^(2) - r_(0)^(2))`
`implies (dr)/(dt) = omega sqrt(r^(2) - r_(0)^(2))`
`implies int (dr)/(sqrtr^(2)- r_(0)^(2)) = intomega dt `
put `r = r_(0) ` sec `theta`
`therefore " " dr = r_(0) "sec" theta tan theta d theta `
`implies " " int(r_(0)sectheta tan theta d theta)/(sqrt(r_(0)^(2) sec^(2)theta - r_(0)^(2))) = int omega dt `
`implies int sec theta d omega = intomega dt `
In `(sec theta + tan theta ) = omega t + C `
at t = 0 , `theta = 0`
`implies C = 0`
`implies "In" (sec theta + tan theta ) = omega t `
`implies sec theta + tan theta = e^(omega t ) " " ....(i)`
using the identify
`sec^(2) theta - tan^(2) theta = 1`
`sec theta - tan theta = e^(- omega t) " " .....(ii)`
from Eqns . (i) and (ii)
`sec theta = (1)/(2) (e^(omega t) + e^(- omega t ) )`
`implies " " (r)/(r_(0)) = (1)/(2) [e^(omega t) + e^(- omega t )]`
`implies " " r = (r_(0))/(2) {e^(omega t) + e^(- omega t )}`
`implies " " r= -(R)/(4) [e^(omega t) + e^(- omega t )]`