Irodov Problem 1.73

Let be the tension in string connected to mass and let T be the tension in the string connected to masses and . Further let be the acceleration of the mass and let w be the acceleration of the mass relative to the moving pulley B (holding masses and ) directed towards pulley B. The net acceleration of the masses and with respect to a stationary frame are and respectively.

Forces on mass : Theres only one force on this mass the tension in the string over pulley A. Thus, the dynamics of the mass are given as,





Forces on mass : There are two forces acting on this mass, i) the tension in the string T pulling it up and the force of gravity pulling it downwards. The net acceleration of this mass is towards the floor. Thus,



Forces on mass : There are two forces acting on this mass, i) the tension in the string T pulling it up and the force of gravity pulling it downwards. The net acceleration of this mass is is towards the floor. Thus,



Forces on the Pulley B: There are three forces acting on pulley B, i) tension pulling it towards pulley A and ii) tensions T on each of the two parts of the string. The mass of the pulley is 0 and its acceleration is acting downwards. Thus, we have,



Now we have all the information to solve for all the unknowns as follows,



From (1), (4) we have,




From (2) and (5) we have,








Similarly, from (3) and (5) we have,








From (6) and (7) we can now find w as,





Now we can use (6) and (8) to find T as,




From (9) and (5) we have,




Finally we can determine the accelerations of both the masses and respectively as,