A mass of 10 kg is suspended using two strings. One string makes an angle of `60^(@)` with the vertical. What should be the angle made by the other string so that the tension in the string will be minimum? Find the tension in each wire.

Explanation

As the mass is in equilibrium, from the force diagram Fig. we get,
`T_(1)costheta_(1)+T_(2)costheta_(2)=mg`
or,`T_(1)costheta_(1)=mg-T_(2)costheta_(2)" "cdots(1)`
Also `T_(1)sintheta_(1)= T_(2)sintheta_(2)" "cdots (2)`
From (2) and (1) we get,
`(sintheta_(1))/(costheta_(1))=(T_(2)sintheta_(2))/(mg-T_(2)costheta_(2))`
or, mg`sintheta_(1)=T_(2)(sintheta_(1)costheta_(2)costheta_(1)sintheta_(2))`
`:. T_(2)= (mgsintheta_(1))/(sin(theta_(1)+theta_(2)))" "cdots(3)`
As m, g and `theta_(1)` are constants the minimum value of `T_(2)` corresponds to the maximum value of sin `(theta_(1)+theta_(2))` i.e., sin`(theta_(1)+theta_(2))=1 = sin90^(@)`
`:. theta_(1)+theta_(2)=90^(@) " ""or",theta_(2)=90^(@)-theta_(1)=90^(@)-60^(@)=30^(@)`
From equation (3) we get,
`T_(2)=(10xxg sin60^(@))/(1)=84.87 `N
From equation (2) we get,
`T_(1)= (T_(2)sintheta_(2))/(sintheta_(1))=(5sqrt3xx9.8xxsin30^(@))/(sin60^(@))=49` N.