Question

The dynamics equations of motion for rigid body motion is a plane is given by:

ΣF = ma       (translational acceleration)
ΣM_CM = I_CMα   (rotational acceleration)

where M_CM is the moment about the Center of Mass, I_CM is moment of inertia about the Center of Mass, and α is the angular acceleration.

Prove that for static or quasi-static analysis, the sum of moments can be taken about any point, as shown below

ΣF = 0       (translational equilibrium)
ΣM_a = 0   (rotational equilibrium)

where M_a is the moment about any point.

Answer 1

In static situations

net force on the system is zero. Hence this is also called translational equilibrium.

net torque about center of mass is zero. That is

Now consider the forces and moments about any other point A

Forces do not change That is net force is still zero.

net torque (sum of moments) about a point is

First sum is sum of moments about center of mass which is zero and second term is zero because net force is zero.

Hence in static situations moments can be taken about any point