Lecture 07 - Lecture 07 * MOMENT ABOUT AN AXIS * MOMENT OF...

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Statics: Lecture Notes for Sections 4.5-4.7 1 Lecture 07 * MOMENT ABOUT AN AXIS * MOMENT OF A COUPLE * EQUIVALENT SYSTEMS Section 4.5-4.7 Ehab Zalok MOMENT ABOUT AN AXIS (Section 4.5) Objectives : Students will be able to determine the moment of a force about an axis using a) scalar analysis, and b) vector analysis. 2
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Statics: Lecture Notes for Sections 4.5-4.7 2 READING QUIZ 1. When determining the moment of a force about a specified axis, the axis must be along _____________. A) the x axis B) the y axis C) the z axis D) any line in 3-D space E) any line in the x-y plane 2. The triple scalar product u • ( r F ) results in A) a scalar quantity ( + or - ) B) a vector quantity 3 A) a scalar quantity ( + or ). B) a vector quantity. C) zero. D) a unit vector. E) an imaginary number. APPLICATIONS 4 With the force F , a person is creating the moment M A . What portion of M A is used in turning the socket? The force F is creating the moment M O . How much of M O acts to unscrew the pipe?
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Statics: Lecture Notes for Sections 4.5-4.7 3 SCALAR ANALYSIS Recall that the moment of a force about any point A is M A = F d A where d A is the perpendicular (or shortest) distance from the point to the force’s line of action . This concept can be extended to find the moment of a force about an axis. 5 In the figure above, the moment about the y-axis would be M y = 20 (0 . 3) = 6 N·m. However, this calculation is not always trivial and vector analysis may be preferable. VECTOR ANALYSIS Our goal is to find the moment of F ( th t d t tt th bd) (the tendency to rotate the body) about the axis a’-a. First compute the moment of F about any arbitrary point O 6 First compute the moment of about any arbitrary point O that lies on the a’a axis using the cross product. M O = r F Now, find the component of M O along the axis a’-a using the dot product. M a = u a M O
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Statics: Lecture Notes for Sections 4.5-4.7 4 VECTOR ANALYSIS (continued) M a can also be obtained as The above equation is also called the triple scalar product. In the this equation, 7 u a represents the unit vector along the axis a’-a axis, r is the position vector from any point on the a’-a axis to any point A on the line of action of the force , and F is the force vector. EXAMPLE Given: A force is applied to the tool to open a gas valve. Find: The magnitude of the A moment of this force about the z axis of the value. Plan : 1) We need to use M = u • (r F). B 8 1) We need to use z u (r 2) Note that u = 1 k . 3) The vector r is the position vector from A to B.
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Lecture 07 - Lecture 07 * MOMENT ABOUT AN AXIS * MOMENT OF...

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