Forces and Moments: Part 3

Moment about an axis:

Sometimes the moment about a point is known and you are supposed to calculate its component about an axis. To find the moment, consider the dot product of M_o and unit vector along axis a:

O: any point on a-a

You can also find the tangent force F_θ and then r x F_θ is the moment about aa:

Let us consider the following example:

Force F causes a moment MO about point O. What is the component of MO along axis oy (My)? Solution:

There are 2 methods to find M_y

1. Scalar Analysis
2. Vector Analysis

(1) Scalar Analysis (first way):

M_O=(20)(0.5)=10 N.m

M_O tends to turn the pipe around axis ob. The component of M_O along the y-axis, M_y, tends to unscrew the pipe from the flange at O.

Thus it is important to know its value.

M_y = (3/5)(10) = 6 N.m

Scalar Analysis (second way):

To find M_y directly (not form M_O) it is necessary to determine the moment-arm, knowing that the distance from F to the y-axis is 0.3m:

M_y = (20)(0.3) = 6 N.m

In general, If the line of action of a force F is perpendicular to any specific axis aa thus:

M_a = F .d_a

(2) Vector Analysis:

First, use the cross product formula to calculate the moment about O:

M_O = r_A × F

M_O = (0.3i + 0.4j) × (-20k)

M_O = {-8i + 6j} N.m

Then use the dot product of M_O and the unit vector along y-axis to get M_y:

M_y = M_O . u_a

M_y = (-8i + 6j). (j)

M_y = 6 N.m