Moments of the forces Mo = F x d A moment is a turning force. Lecture 3 A moment is a turning force. A turning force happens when a force is applied to something which has a pivot. The moment of a force: with respect to a line perpendicular to a plane containing the force is: the product of the force and the perpendicular distance from the force to the line or moment axis. Mo = F x d 90o F d

Moment of Force F around point O Lecture 3 The moment of a force about a point or an axis (MO), provides a measure of the tendency of the force to cause a body to rotate about the point or axis. Oz ┴ plane xy in which Fx lies. Fx causes the pipe to turn about the z-axis. Fx causes a moment about the z-axis = (Mo)z Fy passes through O. Fy does not cause the pipe to turn because the line of action of the force passes through O. NO moment!!

Moment of Force F around point O Lecture 3 2 - D Moment Ox ┴ plane zy in which Fz lies. Fz causes the pipe to turn about the x-axis. Fz causes a moment about the x-axis (Mo)x . Fz is parallel to z- axis, the moment of Fz about z-axis is equal to zero. Fz tends to translate the pipe in z - direction 3 - D Moment If the force does not lie in a plane perpendicular to the moment axis, it may be resolved into two or three, one of them being parallel to the moment axis and the others lying in a plane perpendicular to the axis of moment or rotation. X Z Y F Fx Fy Fz A B C D E G H

Moment (Scalar or Vector?!) Lecture 3 Moment (Scalar or Vector?!) Moment is a vector and the direction is defined using right hand rule. Keep your fingers along the line of action of the force, and your thumb will be along the axis of the moment. M F r M F r Magnitude of the Moment Direction of Moment The direction of Mo would be specified by using the right hand rule. Counter Clockwise (CCW) is out of the page Clockwise (CW) is into the page.

Calculating the moment in 2-D using components Lecture 3 Calculating the moment in 2-D using components Select a positive direction (CCW or CW). Calculate each moment and add them, using the proper sign for each term. Always remember to write the unit of moment which is (N.m or lb.f). Example 1: In the following figure, calculate the moment about the point O. We choose the CCW as positive direction for moment, Moment of component of F along x about O is Fx times the perpendicular distance from O (or d1), which is clockwise, so it is ( - Fx . d1 ) Moment of component of F along y about O is Fy times the perpendicular distance from O (or d2), which is counter clockwise, so it is (Fy . d2 ) Moments add together as vectors, so the total moment is: F

Calculating the moment in 2-D using components Lecture 3 Calculating the moment in 2-D using components Example 2: In the following figure, if ( q ) is 60 degrees and r is 30 mm and F is 6 N, what is the magnitude of the moment about O. We choose the CCW as positive direction for moment, Component of F along r (or F1 = F cos q ) produces no moment, since it passes from point O. Component of F perpendicular to r (or F2 = F sine q ) produces the moment. So the total moment of F about O is: Remember: The moment about O is also calculated using the magnitude of force F times perpendicular distance from O to the line of action of F which is d : F1 q F2 F F2 Note: Moving a force along its line of action does not change its moment!

Moment of a Couple Couple Moment rB - rA = r (as a vector) M = r x F Lecture 3 Moment of a Couple A couple is defined as: Two parallel forces [ F and ( – F ) ] have Same magnitude, Opposite direction and separated by a perpendicular distance ( d ). -F d F Couple Moment A moment produced by a couple is called a Couple Moment. But: rB - rA = r (as a vector) Thus: M = r x F So a couple moment is a free vector which can act at any point and depends only on r, not on rA and rB.

Moment of a Couple Lecture 3 Remember: Scalar Formulation: Magnitude: M = F . d Direction and sense using right-hand rule. Vector Formulation: Magnitude: M = d x F Note: The moment of a couple does not depend on the point one takes the moment about. In other words, a moment of a couple is the same about all points in space. Example 3: The crossbar wrench is used to remove a lug nut from the automobile wheel. The mechanic applies a moment couple to the wrench such that his hands are a constant distance apart. Is it necessary that a = b in order to produce the most effective turning of the nut? Explain. Also what is the effect of changing the shaft dimension c in this regard? The forces act in the vertical plane.

Force – Couple System Equivalent Couple Lecture 4 The combination of the force and couple in the shown figure is refered to as a force – couple system. F Equivalent Couple Changing the values of F and d does not change a given couple as long the product Fd remains the same .

Moment of a Couple Lecture 3 Example 3: Solution: Couple moment: Mcouple = F . ( a + b ), The couple moment depends on the total distance between grips. a = b is not a necessary condition to produce the most effective turning of the nut. Changing the dimension c has no effect on turning the nut. Example 4: Determine the couple moment of the two couples that act on the pipe assembly about ( X, Y and Z) axes. The distance from A to B is d = 400 mm. Express the result as a Cartesian vector. 50 N - 50 N

Moment of a Couple Lecture 3 Example 3: Solution: Lets check three different views and see what forces are causing moments about x, y and z axes: Scalar analysis: Summing moments about ( X, Y and Z ) axes.