To determine the necessary and sufficient conditions for a rigid body to be in equilibrium, we need to consider both linear and rotational aspects of motion.
1. Equilibrium in Linear Motion:
For a rigid body to be in translational equilibrium, the net force acting on the body must be zero. This ensures there is no acceleration according to Newton's First Law of Motion. Mathematically, this can be written as:
[tex]\[\sum \vec{F} = 0\][/tex]
This equation signifies that all the forces acting on the body must cancel each other out.
2. Equilibrium in Rotational Motion:
For a rigid body to be in rotational equilibrium, the net torque acting on the body must be zero. This ensures there is no angular acceleration around any axis. Mathematically, this can be represented as:
[tex]\[\sum \tau = 0\][/tex]
This equation signifies that all the torques acting on the body must cancel each other out.
Therefore, combining these two conditions, the necessary and sufficient condition for a rigid body to be in equilibrium is that both the sum of all forces and the sum of all torques acting on the body must be zero.
Given these conditions, the correct answer for the question is:
D. All forces and the sum of all torque be zero
This combination ensures both linear and rotational equilibrium, thereby meeting the criteria for a rigid body to be in complete equilibrium.
