Irodov 1.123

 We need to calculate the minimum constant force which can move m2. Along with the solution of problem 1.123, I have provided solutions for modified versions of this question where spring is replaced by string and also the minimum force needed to move 3rd block when 3 blocks are connected by springs. I hope this makes your concepts clear. 






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IE Irodov 1.267

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Simultaneous rotation is happening here about 2 mutually perpendicular axes. Rotational kinetic energy is found about both the axes and their algebraic sum gives total KE.  Aliter:  In the alternative method,  KE of centre of mass(translational) about the external axis is added with KE about centre of mass (which is purely rotational). The aggregate is then added with rotational KE about it's own axis to give total KE. 
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