A rigid rod of length 2L is acted upon by some forces. All forces labelled F have the same magnitude. Which cases have a non-zero net torque acting on the rod about its centre?

 

 

 

By sign convention anticlockwise moment (or torque) is taken as position while clockwise moment (or torque) is taken as negative.
In case I, net torque about its centre is

$\mathrm{\tau } = \mathrm{F}\times \mathrm{L} + \mathrm{F}\times \mathrm{L} = 2\mathrm{FL}$

In case II, net torque about its centre is

$\mathrm{\tau } = \mathrm{F}\times \mathrm{L}+\mathrm{F}\times \frac{\mathrm{L}}{2}+\mathrm{F}\times \mathrm{L} = \frac{5}{2}\mathrm{FL}$

In case III, net torque about its centre is

$\mathrm{\tau } = -\mathrm{F}\times \mathrm{L} + \mathrm{F}\times \frac{\mathrm{L}}{2}-\mathrm{F}\times \frac{\mathrm{L}}{2}+\mathrm{F}\times \mathrm{L} = 0$