An infinite number of masses are placed on a frictionless table and they are connected via massless strings. Their masses follow the sequence, m, m 2 , m 6 , . . . m n ! , . . . and they are further connected to a mass m that hangs over a massless pulley. The acceleration of the hanging mass is

Step-by-step solution

The given situation is shown in the following figure Effective mass of the system of n masses placed on the table, M=m+ m 2 + m 6 + + m n ! =m (1+ 1 2 + 1 6 + + 1 n ! )=m ( 1 1 ! + 1 2 ! + 1 3 ! + + 1 n ! )=m (1+ 1 1 ! + 1 2 ! + 1 3 ! + + 1 n ! -1 )M=m(e-1) [ e=1+ 1 1 ! + 1 2 ! + 1 3 ! + + 1 n ! ] The given system of masses can be redrawn as If a be the acceleration of system of masses, then according to Newton's second law of motion, T=M a T=m(e-1) a (i) and m g-T=m a T=m g-m aT=m(g-a) (ii) From Eq. (i) and Eq. (i