A rigid body is sai to be in mechanical equilibrium, If both the linear momentum and angular momentum are not changing with time, the body has neither linear acceleration nor angular acceleration.
(i) Translational equilibrium :
If the total force means the vector sum of the forces on the rigid body is zero, then
`vec(F_(1))+vec(F_(2))+....+vec(F_(n))=0" "....(1)`
`therefore underset(i=1)overset(n)sumvec(F_(i))=0` where `i=1,2,3,...,n`
If the total force on the body is zero, then the total linear momentum of the body does not change with time.
Equation (1) gives the condition for the translational equilibrium of the rigid body.
(ii) Considition for the rotational equilibrium :
If the total torque means the vector sum of the torques on the rigid body is zero, then
`vec(tau_(1))+vec(tau_(2))+...vec(tau_(n))=0" "....(2)`
`therefore underset(i=1)overset(n)sum vec(tau_(i))=0` where `i=1,2,3,....,n`
gives the condition for the rotational equilibrium rigid of the body.
If the total torque on the rigid body is zero, the total angular momentum of the body does not change with time (Remains constant)
The rotational equilibrium condition is independent of the location of the origin.
Equation (1) is equivalent to three scalar equations as under, they corresponds to
`underset(i=1)overset(n)sumvec(F_(ix))=0,underset(i=1)overset(n)sumvec(F_(iy))=0 and underset(i=1)overset(n)sumvec(F_(iz))=0" "......(3)`
where `i=1,2,3,.....,n and F_(ix), F_(iy) and F_(iz)` are respectively the X, Y and Z components of the forces `vec(F_(i))`.
Equation (2) is equivalent to three scalar equations
`underset(i=1)overset(n)sumvec(tau_(ix))=0,underset(i=1)overset(n)sumvec(tau_(iy))=0 and underset(i=1)overset(n)sumvec(tau_(iz))=0" "......(4)`
where `i=1,2,3,....,n and tau_(ix),tau_(iy) and tau_(iz)` are respectively the X, Y and Z component of the torque `vec(tau_(i))`.
There are six independent and in scalar form conditions to be satisfied for mechanical equilibrium of a rigid body.
The sum of the components of the forces along any two perpendicular axes in the plane must be zero.
The sum of the components of the torques along any axis perpendicular to the plane of the forces must be zero.
