Originally Posted by **werehk** the choices are mgcos$\displaystyle \theta$
mgsin$\displaystyle \theta$
mg sin$\displaystyle \theta$cos$\displaystyle \theta$
mgsin$\displaystyle ^2\theta$
What is a systematic approach for the problem |

The most general approach to this would be to apply a force in an arbitrary direction, then use Calculus to find a minimum.

However we have a shortcut available. The minimum force is going to be in such a direction so as not to increase the tension, which would be "wasted" force, and also to not reduce the tension, which is helping hold the mass up. So we want the force to be directed perpendicular to the tension. The rest can be easily worked out from there.

-Dan