# Equilibrium of a rigid body

**1. First condition**

Newton’s Second Law states that if the resultant force on an object (an object considered as a single particle) is not zero then the object will move with constant acceleration, where the direction of the object’s motion = the direction of the total force. If the resultant force is zero then the object is at rest or moving at a constant speed.

ΣF = m a

When an object is rest or moves at a constant speed, the object does not have acceleration (a). Because acceleration (a) = 0, the equation above changes to:

ΣF = 0

This equation can divide into its components on the x-axis, y-axis, and z-axis

ΣFx = 0 (1)

ΣFy = 0 (2)

ΣFz = 0 (3)

If the forces work in the horizontal direction, we use equation 1. If the forces work only in the vertical direction, then we use equation 2. If the forces work in a plane (two dimensions), then we use equation 1 and equation 2. Conversely, if the forces work in space (three dimensions), then we use equations 1, 2, and 3.

Force is a vector quantity, the force has a magnitude and direction. Referring to Cartesian coordinates (x, y, z) and in accordance with the agreement, if the force is in the direction of the negative x-axis (to the left) or in the direction of the negative y-axis (downward), then the force is negative. Conversely, if the force is in the direction of the positive x-axis (to the right) or in the direction of the positive y-axis (upward), then the force is positive.

Sample 1.

F = Pull force, fg = friction force, N = normal force, w = weight, m = mass, g = acceleration of gravity. The object is rest because of the sum of all the forces acting on it = 0.

Review each force acting on the object.

The force acting in the horizontal direction (x-axis):

ΣFx = 0

F – fg = 0

F = fg

Pull force (F) and friction force (f_{fr}) have the same magnitude but in the opposite direction. The direction of the pull force to the right or towards the positive x-axis (positive value), instead of the direction of the friction force to the left or towards the negative x-axis (negative). Because the magnitude of both forces is the same (indicated by the length of the arrow) and the direction is opposite, the magnitude of these two forces is 0.

The force acting on the vertical component (y-axis):

ΣF_{y }= 0

N – w = 0

N – m g = 0

N = m g

In the vertical component (y-axis), there are weight (w) and normal force (N). The direction of weight is perpendicular to the center of the earth or towards the negative y-axis (negative value), while the direction of the normal force towards the y-axis is positive (positive value). The magnitude of these two forces is the same but the direction is opposite so the two forces are eliminating each other.

The object in the example above is at rest because the total force or the sum of all the forces acting on the object, both on the horizontal axis and the vertical axis = 0.

Sample 2.

The weight and the normal force acting on this object are not drawn because both forces are eliminating each other. On both sides of the object is exerted the force F as shown. Both forces have the same magnitude, but opposite direction. Will object rest?

To help you understand this, place a book on the table. At first, the book was rest because the resultant force acting on the book was zero.

Next, give force to both sides of the book, as shown in the figure. When you act force on both sides of the book, it’s the same as you rotate the book. Of course, the book rotates. The book rotates due to the moment of force generated by the force F. The axis of rotation is located in the middle of the book. If we assume that there is no friction force acting on the object, the resultant moment of force is the number of the moment of force generated by the two forces F. The direction of the object’s rotation is clockwise so that the two moments of force are negative (not eliminating each other).

**2. Second condition**

Based on example 2 above it can be concluded that if the resultant moment of force on an object is not zero (the object is considered a rigid object), then the object will rotate.

Στ = I α

In order for objects not to rotate, the resultant moment of force must be zero. When an object is stationary (not rotating), the object does not have angular acceleration. Because of angular acceleration = 0, the above equation changes to:

Στ = 0