# Course Name - Kennesaw State University Forces and Moments MET 2214 Statics (MET 2214) Prof. S. Nasseri Moments and Forces Part 2 Statics (MET 2214) Prof. S. Nasseri Calculating the moment using rectangular

components The moment of a force F about the axis passing through point O and perpendicular to the plane containing O and F can be expressed using the cross product: Mo = r x F The magnitude of the moment is the area shown below: Statics (MET 2214) Prof. S. Nasseri Calculating the moment using rectangular

components Mo = | r x F | = r F sin r=xi+yj+zk F = Fx i + Fy j + Fz k Statics (MET 2214) Prof. S. Nasseri Resultant Moment Statics (MET 2214) Prof. S. Nasseri

Moments and Forces Part 3 Statics (MET 2214) Prof. S. Nasseri 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 Mo and unit vector along axis a: O: any point on a-a

Statics (MET 2214) Prof. S. Nasseri Moment about an axis You can also find the tangent force F and then r x F is the moment about aa: Statics (MET 2214) Prof. S. Nasseri Example 1

Force F causes a moment MO about point O. What is the component of MO along axis oy (My)? Statics (MET 2214) Prof. S. Nasseri Scalar Analysis There are 2 methods to find My Scalar Analysis Vector Analysis (1) Scalar Analysis (first way): MO = (20)(0.5) = 10 N.m

Imagine that we have found the direction of this moment (shown in figure) MO tends to turn the pipe around axis ob. The component of M O along the y-axis, My, tends to unscrew the pipe from the flange at O. Thus it is important to know its value. My = (3/5)(10) = 6 N.m Statics (MET 2214) Prof. S. Nasseri Scalar Analysis Scalar Analysis (second way):

To find My directly (not form MO) it is necessary to determine the moment-arm, knowing that the distance from F to the y-axis is 0.3m: My = (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: Ma = F .da Statics (MET 2214) Prof. S. Nasseri Vector Analysis First, use the cross product formula to calculate the

moment about O: MO = rA F MO = (0.3i + 0.4j) (-20k) MO = {-8i + 6j} N.m Then use the dot product of MO and the unit vector along y-axis to get My: My = MO . ua My = (-8i + 6j). (j) My = 6 N.m Statics (MET 2214) Prof. S. Nasseri Vector Analysis

We can always combine the two previous equations that you saw into one: MO = rA F yielding a My = (rA F) . ua scalar My = MO . Ua Or My = ua . (rA F) Which is called triple scalar product Statics (MET 2214)

Prof. S. Nasseri Remember the triple scalar product from Math section?!! a ax i a y j az k ax b bx i by j bz k a. b c = bx c c i c j c k cx x y z

Magnitude of moment M about axis aa Vector of moment M about axis aa Statics (MET 2214) Prof. S. Nasseri ay by

cy az bz by cz bz c y ax bx cz bz cx a y bx c y by cx az cy ua x ua y uaz M a ua r F rx

ry rz Fx Fy Fz uax

ua y ua z M a M a ua rx Fx ry Fy rz ua Fz

Example 2 Statics (MET 2214) Prof. S. Nasseri Example 2 Statics (MET 2214) Prof. S. Nasseri Example 2 Statics (MET 2214)

Prof. S. Nasseri