A thin massless thread is wound on a reel of mass 3 kg and moment of inertial 0.6 `kg-m^(3)` the hub radius is `R=10`cm and peripheral radius is `2R=20` cm the reel is placed on a rough table and the friction is enough to prevent slipping. find the acceleration of the centre of reel and of hanging mass of 1 kg.

Explanation

Let , `a_(1) = ` acceleration of centre of mass of reel
`a_(2) = ` acceleration of 1 kg block
`omega =` angular acceleration of reel (clockwise)
T = tension in the string
and f = force of friction
Free body diagram of reel is as shown below :
(only horizontal forces are shown).
Equations of motion are :
`T - f = 3a_(1) " ". . . (i)`
`alpha = (tau)/(l) = (f(2R)-T . R)/(l) = (0 . 2f - 0 . 1 f T )/(0 . 6) = (f)/(3) - (T)/(6) " " . . . (ii)`
Free body diagram of mass is, Equation of motion is,
`10 -T = a_(2) " ". . . (iii)`
For no slipping condition,
` a_(1) = 2 R alpha or a_(1) = 0 2 alpha " ". . . (iv) `
and ` a_(2) = a_(1) - R alpha or a_(2) = a_(1) - 0 1 alpha " ". . . (v) `
Solving the above five equations, we get
` a_(1) = 0 . 27 m..s^(2) and a_(2) = 0 . 135 m//s ^(2)`