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 F→rot experienced by the particle of mass m moving on the rotating disc and the force Fin→ experienced by the particle in an inertial frame of reference is F→$$rot=Fin$$→$$+2mv$$→rot×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 center of the discNow 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 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 ω→$$=$$ω$$k^$$ . A small block of mass m is gently placed in the slot at r→$$=R/2i^$$ at $$t=0$$ and is constrained to move only along the slot The distance r of the block at time t is The net reaction of the disc on the block is

Question

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 F→rot  experienced by the particle of mass m moving on the rotating disc and the force Fin→ experienced by the particle in an inertial frame of reference is  F→$$rot=Fin$$→$$+2mv$$→rot×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 center of the discNow 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 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 ω→$$=$$ω$$k^$$ . A small block of mass m is gently placed in the slot at  r→$$=R/2i^$$ at $$t=0$$  and is constrained to move only along the slot The distance r of the block at time t is The net reaction of the disc on the block is

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From the frame of disc the block is constrained to move radially outward due to centrifugal force.F→$$rkt=m$$[$$($$ω→×r→$$)$$×ω→]$$=mr$$ω$$2i^$$⇒$$a=r$$ω$$2$$⇒$$vdvdr=r$$ω$$2$$⇒∫$$0V$$ $$vdr=$$ω$$2$$∫$$R2r$$ rdr⇒$$v2=$$ω$$2r2$$−$$R24$$⇒$$v=$$ω$$r2-R22Now$$ , $$dr=vdt$$⇒$$dr=$$ω$$r2$$−$$R22dt$$⇒∫$$R2r$$ $$drR2r2$$−$$R22=$$ω∫$$0t$$ dt⇒cosh−$$1$$⁡$$2rR=$$ωt⇒$$2rR=cosh$$⁡$$($$ω$$t)$$⇒$$r=R2e$$ω$$t+e$$−ω$$t2=R4e$$ω$$t+e$$−ωt F→$$rot=F$$→$$in+2mv$$→rot×ω→$$+m($$ω→×r→$$)$$×ω→$$=$$−mω$$2ri^+2mvrot$$ω$$($$−$$j^)+m$$ω$$2ri^=$$−$$2mvrot$$ω$$j^here$$ $$vrot=drdt=$$ω$$R4e$$ωt−e−ωtF→rot $$=$$−mω$$2R2e$$ωt−e−ω$$tj^The$$ reaction force on block due to walls of disc will be −F→rot The net reaction force will beR→$$=R$$→wall $$+R$$→ground $$=m$$ω$$2R2e$$ωt−e−ω$$tj^+mgk^$$

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