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

Here, number of unknowns are five:
`a_(1)=` acceleration of centre of mass of reel
`a_(2)=` acceleration of 1 kg block
`alpha=` angular acceleration of reel (clockwise)
`T=` tension in the string
`f=` force of friction
Acceleration equations:
Free bod diagram of reel is as shown in figure, (only horizontal forces are shown)
equations of motion are
`T-f=3a_(1)` ..(i)
`alpha=(tau)/(I)=(f(2R)-T.R)/(I)=(0.2f-0.1T)/(0.6)=(f)/(3)-(T)/(6)` ..(ii)

Free body diamgram of mass is
equation of motion is Equation of motion is
`10-T=a_(2)` ..(iii)
contact equations
for no slipping condition `a_(1)=2Ralpha` or `a_(1)=0.2alpha` ..(iv)
and `a_(2)=a_(1)-Ralpha` or `a_(2)=a_(1)-0.1alpha`
solving the above five equations we get
`a_(1)=0.27m//s^(2)` and `a_(2)=0.135m//s^(2)`