What equation means that the vector sum of all the forces act on an objcect in an equiliberium equals zero?

Question

forces cancel

Answer 1

Equilibrium means that the sum of all vectors is zero.

A+B+C=0

It also means that the sum of vectors' components is also zero

Ax+Bx+Cx=0
Ay+By+Cy=0
Az+Bz+Cz=0 ans so on


For the sake of a simplest example let's take
A+B=0 in two dimensions

Ax +Bx=0 or Ax= -Bx
Ay +By=0 or Ay= -By

Answer 2

When several forces act on an object (which is a system of particles), following conditions should be satisfied for the equilibrium of the body. Equilibrium means the object continues in its state of rest if it was stationary or of uniform velocity if it was moving.

Conditions:

( 1 ) The vector sum of all the forces must be zero.

( 2 ) The vector sum of torques due to all the forces acting on the object with respect to some reference point is zero.

If the forces are concurrent, then if first condition is satisfied, the second will be satisfied. So in that case, we need to check only the first condition.

If the forces are not concurrent, then it is necessary that both the conditions are satisfied. If only the first condition is satisfied, then the body will have a pure rotational motion. If only the second condition is satisfied, then the body will have pure translatory motion.