# Engineering Mechanics

ENGINEERING MECHANICS GE 107 Lecture 5 1 Moment of a Force The analysis of a system of forces requires knowledge about the equilibrium For a particle or a system to be in equilibrium, not only the resultant forces but also the resultant of another component known as moment should be equal to zero

When a force is applied to a body , it will produce a tendency for the body to rotate about a point that is not on the line of action of the force. This tendency to rotate is sometimes called a torque but most often it is called the moment of a force or simply the moment 2 Magnitude of Moment For example. consider a wrench used to unscrew the bolt as shown in Fig.1 If a force is applied to the handle of the wrench, it will tend to turn the bolt about point 0 (or the z axis). The magnitude of the moment is directly proportional to the magnitude of F and the perpendicular distance or moment arm d

if the force F is applied at an angle 090. then it will be more difficult to turn the bolt since the moment arm d =dsin will be smaller than d. lf F is applied along the wrench. its moment arm will be zero since the line of action of F will intersect point 0 (the = axis). As a result. the moment of F about 0 is also zero and no turning can occur. The Magnitude of Moment about the axis O is 3 Direction of Moment The direction of Mo is defined by its moment axis, which is perpendicular to the plane that contains the force F and its moment arm d The right-hand rule is used to establish the sense of direction of Mo

According to this rule. the natural curl of the fingers of the right Hand, as they are drawn towards the palm, represent the tendency for rotation caused by the moment. As this action is performed. the thumb of the right hand will give the directional sense of MO as in Fig. a. The moment vector is represented three dimensionally by a curl around an arrow. In two dimensions this vector is represented only by the curl as in Fig. b. in counterclockwise rotation. 4 Resultant Moment For two-dimensional problems. where all the forces lie within the .X-Y plane as in Fig., the resultant moment (M R) O

will be about point 0 (the z axis) It can be determined by the algebraic sum of the moments caused by all the forces in the system. As a convention, we will generally consider positive moment as counter clock wise since they arc directed along the positive z axis (out of the page). Clockwise moments will be negative Doing this, the directional sense of each moment can be represented by a plus or minus sign. Using this sign convention, the resultant moment in Fig. is therefore 5

Example For each case illustrated in Fig. determine the moment of the force about point O. Solution: The line of action of each force is extended as a dashed line in order to establish the moment arm d. 6 Example 1 (Contd.) 7 Example 2 Determine the resultant moment of the four forces acting on the rod as shown in Fig. about point 0

8 9 Effect of Moment Bending Moment Nail removal Cross Product The knowledge of cross product of vectors is necessary to derive the vector formulation of moment The cross product of two vectors A and B yields the vector C, which is written as

C=AxB The magnitude of cross product C is defined as the product of the magnitudes of A and B and the sine of the angle between their tails. Thus. C = AB sin . The direction of Vector C is perpendicular to the plane containing A and B such that C is specified by the right-hand rule: i.e ..curling the fingers of the right hand from vector A (cross) to vector B, the thumb points in the direction of C. as shown in Fig. Knowing both the magnitude and direction of C, we can write C = A xB= (AB sin )uC where the scalar AB sin defines the magnitude of C and the unit vector defines the direction of C. 10 11

Cross Product (Contd..) The cross product is not commutative i.e ., A X B B X A, Rather A X B = - B X A If the cross product is multiplied by a scalar a, it obeys the associative law: a(A X B) = (aA) xB= A X (aB) = (A X B)a The vector cross product ,also obeys the distributive law of addition A X (B + D)=(A X B ) + (A X D) Cross Product (Contd..) Cartesian Vector Formulation The Equation C=AB sin may be used to find the cross product of any pair of Cartesian unit vectors.

For example, to find i X j, the magnitude of the resultant vector is ( i)( j )(sin 90") = (1)( 1)( 1) = 1. Its direction is determined using the right hand rule, As shown in Fig., the resultant vector points in the +k direction. Thus i x j =(1)k. In a similar manner. 12 Cross Product Cartesian Vector Formulation (Contd..) Let us now consider the cross product of two general vectors A and B which are to be expressed in Cartesian Vector form. A B ( Ax i Ay j Az k) ( Bx i B y j Bz k)

Ax Bx (i i) Ax B y (i j ) Ax Bz (i k) Ay Bx ( j i) Ay B y ( j j ) Ay Bz ( j k) Az Bx (k i) Az B y (k j ) Az Bz (k k) Carrying out the cross product we get A B ( Ay Bz Az B y )i ( Ax Bz Az Bx ) j ( Ax B y Ay Bx )k) 13 Cross Product Cartesian Vector Formulation (Contd..) The equation can also written in then matrix form as i

A B Ax Bx j Ay By k Az Bz Hence to find the cross product of any two Cartesian vectors A and B, it is necessary to expand a determinant whose first row of elements consists of the unit vectors i, j and k and whose second and third rows represent the x. y, z components of the two vectors A and B respectively

14 Moment of Force- Vector Formulation 15 The moment of a force F about point O, or actually about the moment axis passing through 0 and perpendicular to the plane containing O and F as shown in Fig. can be expressed using the vector cross product as M O r F

-Where r represents a position vector directed from o to any point on the line of action of F. The magnitude of the cross product is defined from MO=rFsin where the angle is measured between the tails of r and F. Since rsin =d, the moment arm, MO=R(rsin )= Fd The direction and sense of MO can be determined by the right-hand rule as it applies to the cross product. 16 Principle of Transmissibility The cross product operation is often used in three dimensions since the perpendicular distance or moment arm from point O to

the line of action of the force (which is difficult to obtain) is not needed. In other words. we can use any position vector r measured from ofthe force F, as shown point O to any point on the line of action M O r1 F r2 F r3 F in Fig. Since F can be applied at any point along its line of action and still create the moment about point O, then F can be considered a sliding vector This property is called the principle of transmissibility of a force. Cartesian Vector Formulation for Moment

Considering the coordinate axes x,y and z. then position vector r and force F can be expressed Cartesian vectors. Applying cross product equation M O i r F rx F x j

ry F y 17 the as k rz F z where rx , ry, rz represent the x, y, z components of position Vector from point O to any point on the action of the force Fx ,Fy, Fz represent the force components of the

vector. the line of force 18 Resultant Moment of Forces If a body is acted upon by a system of forces as shown in Fig., the resultant moment of the forces about point O can be determined by vector addition of the moment of each force. This resultant can be written symbolically as

M O (r F ) Problem 1 Determine the moment produced by the force F as shown in fig. about point O. Express the result as a Cartesian vector. 19 20 Solution to Problem1 As shown in Fig., either rA. or rB can be used to determine the

moment about point O. The position vectors are r A 12 k and r B 4 i 12 j Force F can be expressed in Cartesian form as F F u AB

{ 0 . 4588 Thus M O rA 4 i 12 j 12 k 2 ( 4 2 12 2 ( 12 ) i 1 . 376 j 1 . 376 k } kN

F [ 0 ( 1 . 376 [ 0 ( 1 . 376 i j 0 0 . 4588 ) 12 ( 1 . 376 0

1 . 376 2 k 12 1 . 376 )] i [ 0 ( 1 . 376 )] k ) 0 ( 0 . 4588

{ 16 . 5 i 5 . 5 j } kN m ) 12 ( 0 . 4588 )] j Problem 2 Two forces act on the rod as shown in Fig. Determine the resultant moment they create about the flange at O. Express the result as a Cartesian vector. 21 Solution to Problem 2 Position vectors for each are to be determined first.

The resultant moment about 0 is therefore 22 Principle of Moments A concept often used in mechanics is the principle of moments, which is sometimes referred to as Varignons Theorem It was originally developed by the French mathematician Varignon (1654- 1722). It states that the moment of the of force about a point is equal to the sum of the moments of components of the force about that point. This theorem can be shown by the distribution law of cross product.

Considering the forces shown in Fig., As Therefore 23 24 Principle of Moments (Contd..) For two dimensional problem, we can use the principle of moments by resolving the force into its rectangular components Then determine the moment using a scalar analysis. as Principle of Moments Contd..)

Example1 To determine the moment of the force in Fig. a about point O. The moment arm d as in Fig. b, can be found from trigonometry. d = (3 m) sin 7S= 2.898 m 25 Principle of Moments Contd..) Example 2 Force F acts at the end of the angle bracket as shown in Fig.a. Determine the moment of the force about point O. The force is resolved into two components as shown in Fig.b., then

26 Principle of Moments Example 2 (Contd..) Alternative Method using Varignons theorem Using a Cartesian vector approach, the force and position vectors are determined The moment is therefore 27 Moment of a Force about a Specified Axis 28

Some times-the moment produced by a force about a specified axis must be determined. For example, suppose the lug nut at a on the car tire as in Fig. needs to be loosened. The force applied to the wrench will create a tendency for the wrench and the nut to rotate about axis passing through 0 But the nut can only rotate about the y axis. Hence only the y component of the moment is needed. and the total moment produced is not important. To determine this component, we can use either a scalar or vector analysis.

Moment of a Force about a Specified Axis (Contd..) Scalar Method To use a scalar analysis in the case of the lug nut as in Fig., Moment arm, the perpendicular distance from the axis to the line of action of the force is dy = d cos . Thus. the moment of F about the y axis is My=Fdy= F(d cos ). According to the right-hand rule. My is directed along the positive y axis as shown in the figure.

In general, about any axis a, the moment is 29 Moment of a Force about a Specified Axis (Contd..) Vector Method To find the moment of force F about the y axis using a vector analysis, we must first determine the moment of the force about the point 0 on they axis by applying Equation Mo = r X F.

The component My along the y axis is the projection of Mo on the y axis. It can be found using equation of dot product M y= j Mo= j . (r X F). where j is the unit vector for the y axis. Let ud be the unit vector along any axis a, then The result can be written in the matrix form as 30

Problem 3 Determine the moment MAB produced by the force F in Fig.a, which tends to rotate the rod about the AB axis. 31 Solution to Problem 3 The vector approach MAB= uB(r X F) can be used to find the moment uB defines the direction of AB axis of the rod r is the position vector directed from any point on AB to any point on the line of action of force The force is

32 Solution to Problem 3 (Contd..) 33 Problem 4 Determine the magnitude and direction of moment of force F about segment OA 34 Solution to Problem 4 To get the force in vector form ,

find the unit vector along OA Find the position vector for force segment OA Hence the moment about OA is 35 Moment of a Couple A couple is defined as two parallel forces that have the same magnitude but opposite directions and are separated by a perpendicular distance d. 36

37 Moment of a Couple (Contd..) For example, imagine that you are driving a car with both hands on the steering wheel and you are making a turn. One hand will push up on the wheel while the other hand pulls down, which causes the steering wheel to rotate Since the resultant force is zero, the only effect of a couple is to produce a rotation or tendency of rotation in a specified direction. This is nothing but a moment or moment of a couple The moment produced by a couple can be

called as Couple Moment. Couple Moment Scalar Formulation The moment of a couple as shown in the Fig. is defined as having a magnitude of Where F is the magnitude of one of the forces and d is the perpendicular distance or moment arm between the forces. The direction and sense of the couple moment are determined by the right hand rule, where the thumb indicates this direction and the fingers are curled with the sense of rotation caused by the couple forces. In all cases, M will act perpendicular to the plane containing the forces.

38 Couple Moment vector Formulation Moment can be determined by finding the sum of the moments of both couple forces about any arbitrary point. For example as in Fig. the position vectors rA and rB are directed from point 0 to points A and B lying on the line of action of - F and F The couple moment determined about 0 is But the position vectors are added as Hence This result indicates that a couple moment is a free vector i.e., it can act at any point since M depends only upon the position vector r directed between the forces.

39 40 Resultant Couple Moment Since couple moments are vectors, their resultant can be determined by vector addition. For example, consider the couple moments M1 and M2 acting on the pipe as shown in the Fig.a. Since each couple moment is a free vector, we can join their tails at any arbitrary point and find the resultant couple moment as shown in Fig.b. MR=M1+M2 If more than two couple moments act on the body, we may generalize this concept and write the vector resultant as

Problem 5 Determine the magnitude and direction of the couple moment acting on the gear as shown in the Figure. (R.C. Hibbeler p151) 41 Solution to Problem 5 Scalar Approach The distance d can be found graphically and substituting in the formula M= Fd , its magnitude can be found

Analytical method to find d is complicated and hence vector approach is used 42 43 Solution to Problem 5 (Contd..) The forces are resolved in to two components The moments of respective resolved components are found about any point (say O) Summing up the moments will give the resultant couple moment caused by the forces

600 sin30N 0.2 m M1 600 sin30N 600 cos30N M 0.2 m This positive result indicates that M has a counter clock wise2 rotational sense and it is directed outward . 600 cos30N perpendicular to the p:lgc.

Problem 6 Replace the two couples acting on the pipe column as shown in the figure by a resultant couple moment. 44 45 Solution to Problem 6 Let the couple moments M1 M2 are produced by the and F2 respectively and

forces F1 Where F1 = 150 N and F2= 125 N is given as M The vector M 1 rAB1 F1 By inspecting the figure, r AB is parallel to y axis and hence equal to 0 .4 j m Similarly F1 is parallel to z axis and therefore 150 k N M 0 .4 j 150 k 60 i N m

1

46 Solution to Problem 6 (Contd..) The vector M2 is computed in the similar as manner M2 rDC F2 By inspecting the figure, rDC is parallel to x axis and hence equal to 0.3 i m

Moment of force F2 is obtained by varignon' s theorem by resolving it in two components , one parallel to z axis and the other parallel to y axis. The resultant moment is thus Simplification of Force and couple system 47

Example 48 Example (Contd.) 49 Problem 7 Find the resultant moment for the system shown 50

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