A small block is connected to one of two identical massless strings of lengtth `16(2)/(3) cm` each with their ends fixed to a verticle rod If the ratio of tensions `T_(1)//T_(2)` is 4 : 1 ,then what will be the angular velocity of the block ? Take `g = 9.8ms^(-2)`

Explanation

For horizontal,
`T_(1) sin 60^(@)+ T_(2) sin 60^(@) = m omega^(2)r = m omega^(2)1sin 60^(@)`

`rArr T_(1) + T_(2)= m omega^(2)1`…(i)
For verticle equlibrium of the block
`T_(1) cos = T_(2) cos 60^(@) + mg`
`rArr T_(1) - T_(2) = (mg)/( cos 60^(@) ) = 2 mg` .....(ii)
Dividing (i) by (ii), `(T_(1) +T_(2))/(T_(1) - T_(2)) = (omega^(2)1)/(2g)`
`rArr ((T_(1))/(T_(2)) + 1)/((T_(1))/(T_(2)) - 1)= (omega^(2)1)/(2g) rArr (4 + 1)/(4 - 1) = (omega^(2)1)/(2g)`
or `omega^(2)=(10g)/(3l) = (10 xx 9.8 xx 3)/(3 xx 50 xx 10^(-2)) = 196`
or `omega = 14 rad s^(-1)`