Moment of a Force

Scalar Representation: Consider force acting at point A. The magnitude of the moment of this force about point B is determined from the scalar product

(1)

 

where F is the magnitude of the force, and d is the perpendicular distance between point B and the line of action of the force. Distance d is commonly referred to as the moment armof the force while point B is called the moment center.

In this case, the moment about point B can be interpreted as the measure of the tendency of force F to cause the body to rotate about B. The key word is tendency. It is not necessary for the body to actually rotate about B for the moment to be created. The sense of the moment is determined based on the right-hand rule. The axis of the moment vector is perpendicular to the plane containing and d.

To calculate the moment of a force using the scalar approach, we must know:
  1. Force magnitude
  2. Location of the moment center
  3. Perpendicular distance from the moment center to the line of action of the force

Example 1

Moment Resultant: In the case of two or multiple forces, the moment resultant is found as

(2)

   

Assuming the counter clockwise direction as positive, the moment resultant about point B is found as

If the answer has a positive sign, then the moment is in fact in the counter clockwise direction. If negative, it is in the clockwise direction. Notice that F3 did not appear in the equation as it has a moment arm of zero with respect to point B.

Principle of Moments: Depending upon the direction of the force vector, it may be difficult to directly measure the moment arm das in Eq. (1). In those situations, we break the force into its individual components, and find the moment as the sum of the moments of individual components as

(3)

Care should be taken to account for the direction of each moment component in Eq. (3). This approach is generally referred to as the Principle of Moments,which simply indicates that the moment of a force is equal to the sum of moments of its components.

Consider force acting along the line passing through points A and B. To find the moment of this force about point C, we need to know its moment arm designated as dc. If this distance is not readily available, then it is easier to apply the principle of moments. Since force is a sliding vector, it can be placed at any point along its line of action. Let's slide it up to point B, and break it into its x,y components. The moment about point C is found as

Notice that both moments are in the clockwise direction. Once we know the moment about C, we can solve for the moment arm (if it was of interest) as

Five-Minute Exercise: Prove that the moment about point C will stay the same if the force is moved to point A instead of B. Let F= 200 lb, a= 3 ft, b= 4 ft, and c= 6 ft.

Example 2

Vector Representation: The moment of a force about an arbitrary moment center "O" can also be described by the vector equation

(4)

where is the position vector measured from the moment center to any point along the line of action of the force vector . The directional sense of the moment is found by first aligning the position and the force vectors tail to tail, then curling the fingers in right hand from to . The axis of the moment vector passes through the moment center and is perpendicular to the plane containing and . The magnitude of moment is measured in units of force times length (e.g., lb·in or N·m).

To calculate the moment of a force using the vector approach, we must know:

  1. Force vector
  2. Location of the moment center
  3. Position vector (measured from the moment center to any point along the line of action of the force vector)

The following example describes the calculation of moment using the vector approach.

Example 3 (LiveMath)

It is important to note that the position vector can be measured from the moment center to any point along the line of action of as shown in the figure. The resulting cross products in all cases will be equal. That is

The reason all cross products in the above equation give the same answer is because

and

where q is the angle between the tails of and .

Therefore, if we slide the force vector to any location along its line of action, its moment with respect to any arbitrary point or moment center will remain unchanged. This attribute is generally known as the principle of transmissibility. This fact is demonstrated in the following example.

Example 4 (LiveMath)

In the case of two or more forces, Eq. (4) is expanded to

(5)

where N is the number of forces and represents the position vector measured from the moment center to any point on the line of action of . The following example shows the calculation of moment due to multiple forces.

Example 5 (LiveMath)

Moment about an Axis: In solving problems in statics, we sometimes need to determine the moment of a force about a particular axis. If the line of action of the force is perpendicular to the axis of interest, we then find the moment as

(6)
 

where dab is the moment arm of the force with respect to the axis ab. The direction of Mab follows the right-hand rule. In applying Eq. (6), we must be aware of two commonly encountered situations. They are

  1. When the force and the axis are parallel, the resulting moment is zero.
  2. When the force passes through the axis, the resulting moment is zero.

When it is difficult to find dab, we first calculate the moment about an arbitrary point (moment center) that falls on line ab, then find its projection along line ab. In vector notation, Mab, representing the projection of the moment vector along line ab, is expressed as

(7)
 

where is the unit vector along line ab and is the position vector from the moment center on line ab to any arbitrary point along the line of action of . If Mab is found to be negative, then its direction is opposite to that defined by .

The moment about line ab can be expressed in vector form by simply multiplying its magnitude times the unit vector in the same direction.

(8)

   

Example 5 (LiveMath)

Force Couples: A force couple consists of two parallel forces of equal magnitude and opposite direction separated by a finite distance. The net result is a couple moment of magnitude M = Fd. It is important to realize that a couple moment is a free vector meaning that it can be shown at any point on the body as long as its magnitude and direction are not altered.

=
=

Example 6 (LiveMath)

Test Your Knowledge

Force couples appear in many problems, and they represent a practical way for applying moments on a body. For example, when one uses a cross wrench to loosen or tighten the lug nuts on a wheel of a car, the applied forces at the opposite ends of the cross wrench form a couple. Also by having both hands on the steering wheel of a car, we can apply a couple to turn the wheel.