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1)

A frame of the 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  experienced 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 center of the disc

 

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Now, consider a smooth slot along a diameter of a disc of radius R  rotating counter-clockwise with a constant angular speed ω about its vertical axis through its centre. We assign a coordinate system with the origin at the centre of the disc, the X-axis along the slot, the Y-axis perpendicular to the slot and Z-axis along the rotation axis((ω=ωˆk). A small block of mass m is gently placed in the slot at r=(R2)ˆi at t=0 and is constrained to move only along the slot

The net reaction  of the disc  on the block is


A) mω2Rsinwt˜jmg˜k

B) 12mω2R(eωteωt)ˆj+mgˆk

C) 12mω2R(e2ωte2ωt)˜j+mg˜k

D) mω2Rcosωt˜jmg˜k

Answer:

Option B

Explanation:

1222021613_bbb.JPG

Frot=Fin+2m(vrotˆi)×ωˆk+m(ωˆk×rˆi)×ωˆk

mrω2ˆi=Fin+2mvrotω(^j)+mω2rˆj

Fin=2mvrωˆj

r=R4[eωt+eωt]

 drdt=vr=R4[ωeωtωeωt]

Fin=2mRω4[eωteωt]wˆj

 Fin=mRω22[eωteωt]ˆj

Also , reaction  is due to disc surface , then

 Freaction=mRω22[eωteωt]ˆj+mgˆk