Third System of Pulleys:

Figure shows the third system of pulleys. The frame of the top most pulleys is linked to the fixed support, & other pulleys are arranged in such manner that they may move up and downs illustrates in the figure.

Letting free body diagram

T₁  + T₂  + T₃  + T₄  = W

If there is no friction, and treating the system to be ideal, we get:

                              T₁ = P

Letting the free body diagram of the lower most pulley,

T₂  = 2T₁  = 2P

Likewise, T₃  = 2T₂  = 2² P

and T₄  = 2T₃  = 2³ P

Therefore,  (2³  + 2²  + 2 + 1) P = W

or         W/ P = 1 + 2 + 2² + 2³

Since the system is ideal,

V R = W /P = 1 + 2 + 2² + 2³

which may be written as = 2⁴  - 1 .

If there are 'n' pulleys, we may write :

V R = 2ⁿ   - 1