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

Answer

B

Explanation

It is given that X-axis is along the groove and Y-axis is perpendicular to the groove. Z-axis is perpendicular to the plane of the disc.
Above described frame is assumed to be fixed with the axis hence it rotates along with mg the disc. We can understand that in this rotating frame of reference particle is moving along X-axis only. Hence position vector, velocity vector and acceleration vector are directed along the X-axis only.
`vec r= r hati, vecv_(rot)=(dr//dt) hati, vec omega= omega vec k and vec F_(rot)= m(d^92)r)/(dt^(2)) hati N_(1), N_(2)` and mg shown in figure are real forces which act in inertial frame. Hence we can write the following:
`vecF_("in")N_(1) hatj+(N_(2)- mg) hatk " "...(i)`
Force acting on the particle with respect to rotating frame of reference is given as follows:
`vec F_(rot)= vecF_("in")+2m(vecr_(rot)xx vec omega)+m(vec omegaxx vecr)xx vec omega " "...(ii)`
Substituting all values in equation (ii) we get the following:
`m(d^(2) r)/(dt^(2)) hati[N_(1) hatj+(N_(2)- mg) hatk]+2m((dr)/(dt) hatixx omega hatk)+ m(omega hatk xx r hati )xx omega hatk`
`rArr m(d^(2)r)/(dt^(2)) hati= m omega^(2) r hati+(N_(1)-2m omega (dr)/(dt))hatj+ (N_(2)-mg) hatk`
On comparing the individual components on both the sides of the equation we get
`(d^(2)r)/(dt^(2))= omega^(2) r" "...(iii)`
`N_(1)=2m omega (dr)/(dt) " "...(iv)`
`N_(2)= mg " "...(v)`
We can see that value ofr given in option (b) satisfies our equation (iv) and also it satisfies the boundary condition of t=0, R/2. Hence option (b) is correct.