Balancing conditions of a rigid body

For a rigid body to be in balance, besides not moving, this body cannot rotate. So you need to satisfy two conditions:

  1. The resultant of the forces applied on its center of mass must be null (does not move or move with constant velocity).
  2. The resultant Force Moments applied to the body must be null (does not rotate or rotates at constant angular velocity).

Having both conditions met, any body can be in balance, like this pen:


(1) In a circus, a 65kg acrobat is on a uniform 1.2m trampoline, the mass of the trampoline is 10kg. The distance between the base and the acrobat is 1m. Another circus member pulls a rope attached to the other end of the trampoline, which is 10cm from the base. What force does it have to do to bring the system into balance.

Since the trampoline is uniform, its center of mass is exactly in its middle, which is located at a distance of 0.5m from the base. So, considering each force:

For the second equilibrium condition: