In the figure shown all pulleys are ideal and small. The central pulleys is 2 m away from both ends. What is the length (in m) of the string ? The system is in equilibrium

Options

A. `8/(sqrt(3))`
B. `4/(sqrt(4))[1+2/(sqrt(5))]`
C. `4/(sqrt(3))[1+2/(sqrt(3))]`
D. `2/(sqrt(3))+4`

Answer

B

Explanation

All pulley are smooth so tension in sring is same. At all points
For horizontal equilibrium at `P_(1)`
`T sin60^(@)`
For vertical equilibrium at `P_(3)`
`2T cos 60^(@) =20`
T=20 N
for vertical equilibrium at `P_(3)`
`2Txxcos theta=10`
`2xx20xxcos theta=10`
`rArr cos theta=1/4 rArr sin theta=(sqrt(15))/4`
`rArr ` length of string `=2 cosec 60^(@) +2 cosec theta`
`2xx2/(sqrt(3)) +2xx4/(sqrt(15)) =4/(sqrt(15))=4/(sqrt(3))(1+2/(sqrt(5)))`