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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 ω is an example of a non-inertial frame of reference. The relationship between the force Frot experience by a particle of mass m moving on the rotating disc and the force Fin experienced by the particle in an inertial frame of reference is
Frot=Fin+2m(vrot×ω)+m(ω×r)×ω,

where vrot is the velocity of the particle in the rotating frame of reference and 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 counterclockwise with a constant angular speed ω about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the xaxis along the slot, the y axis perpendicular to the slot and the z axis along the rotation axis ω=ωk. A small block of mass m is gently placed in the slot at r=(R/2)i at t=0 and is constrained to move only along the slot.


The net reaction of the disc on the block is

A
mω2Rsinωt^jmg^k
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B
12mω2R(eωteωt)^j+mg^k
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C
12mω2R(e2ωte2ωt)^j+mg^k
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D
mω2Rcosωt^jmg^k
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Solution

The correct option is B 12mω2R(eωteωt)^j+mg^k
Frot=Fin+2m(vrot^i×ω^k)+m(ω^k×r^i)×ω^k
Since Frot will be in +ve x-direction i.e along ^i in the rotating frame of reference and (ω^k×r^i)×ω^k will also be in the same direction whereas Fin will be in direction perpendicular to x-direction since the reactions acting on the particle due to circular disc and weight of the particle will be in y- direction and z-direction respectively .
Hence,
0=Fin+2mvrotω(^j)Fin=2mvrotω^jr=R4(eωt+eωt)vr=drdt=Rω4(eωteωt)Fin=2mω2R4(eωteωt)^j
There wil be a reaction due to the weight of the particle in +ve z-direction given by mg^k
Hence net reaction force on the particle will be
Freaction=mω2R2(eωteωt)^j+mg^k

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