Moment of Force about a Line in Space

Moment of Force about a Line in Space

The magnitude of moment of force about a line can be obtained by first finding moment of force about any point on the line and then finding its projection of that moment on that line.

Consider a force as shown in Fig. 3.12.1. To find its moment about a line AB, first find its moment about a point C on the line AB.

The projection of on the line AB is the magnitude of moment about the line AB i.e.,

The sense of rotation of moment is obtained by observing the moment from B towards A. Moment of force about a line is zero when i) force is parallel to that line or ii) force intersects that line.

Solved Examples for Understanding

Example 3.12.1 

Find the resultant moment of the force system in Fig. 3.12.2 acting on a rectangular plate about a) X-axis b) Y axis c) Z-axis.

Solution: 

When forces are all parallel to one of the axis, moment about coordinate axis can be obtained by looking towards the origin along that axis and calculating moment. Treat anticlockwise moment positive.

⸫ Mx = 100 × 5 + 50 × 2.5 – 300 × 5 = 1875 Nm

 (Negative sign means clockwise moment)

My = 0 as all forces are parallel to Y-axis.

Mz = 200 × 4 – 50 × 4 + 300 × 8 = 3000 Nm

Example 3.12.2 

Determine the moment of force F shown in Fig. 3.12.3 about the line OA.

Solution : 

The required coordinates are A (6, 3, 0), B (4, − 1, 4), C (− 1, 6, 1)

The unit vector along OA is

Example 3.12.3 

A force acts at point C(1, 1, 1). Determine the C(1,1,1). moment of the force about line passing through A (2, 1, 2) and B(4,3,2)

Solution: 

Example 3.12.4 

If the magnitude of moment of about the line AB is 1000 Nm, determine the magnitude of . Refer Fig. 3.12.4.

Solution : 

Example 3.12.5 

Find the moment of the 10 kN force about an axis going between points D and C

Moment of F = 10 kN about line DC is

The co-ordinates of required points are

A (1, 0, 0), B (0, 2, 1), C (0, 2, 0) and D (1, 2, 1)