Consider an arrangement shown in the figure. The pulley P is frictionless and the threads are massless. The mass of the spools is m and moment of inertia of the spool is 1/2 mR^(2). The mass of the disc of radius R is also m. The surface below the spool is rough to ensure pure rolling of spool. the mass of the block is m and the surface below the block is smooth. find the initial acceleration of the block when the system is released from rest.

Consider an arrangement shown in the figure. The pulley P is frictionl

1.	Consider an arrangement shown in the figure. The pulley P is frictionless and the threads are massless. The mass of the spools is m and moment of inertia of the spool is 1/2 mR^(2). The mass of the disc of radius R is also m. The surface below the spool is rough to ensure pure rolling of spool. the mass of the block is m and the surface below the block is smooth. find the initial acceleration of the block when the system is released from rest.
Answer» `4/37 g` `2/37 g` `8/37 g` `10/37 g` Solution :SUPPOSE the acceleration of the block `B` is a, acceleration of disc is `a_(2)` and the acceleration of centre ofmass of spool is `alpha_(1)`. Also suppose the angular accelerations of spool and disc are `alpha` and `alpha_(2)` respectively. Now `alpha=alpha_(1)(R-R/2)` `rArralpha=(alpha_(1)R)/2....(1), alpha_(1)=alpha_(1)R.....(2)` `alpha_(2)=alpha_(1)(2R)+alpha_(2)(R).......(3)` For the block `ma=T_(1)......(4)` For the spool `malpha_(1)=T_(2)-T_(1)-F.....(5)` `((mR^(2))/2)alpha_(1)=T_(2)R+T_(1)R+fR.....(6)` For the disc `ma_(2)=mg-T_(2)` .....(7) `(1/2mR^(2))alpha^(2)=T_(2)R......(8)` For these equations, we get `a=4/37 g`