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.

Options

A. `4/37 g`
B. `2/37 g`
C. `8/37 g`
D. `10/37 g`

Answer

A

Explanation

Suppose the acceleration of the block `B` is a, acceleration of disc is `a_(2)` and the acceleration of centre of mass 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`