Chapter 4 Moment - the tendency of a force to rotate an object
Finding the moment - 2D Scalar Formulation Magnitude of force Mo = F d Rotation is clockwise or counter clockwise Moment about 0 Perpendicular distance between LOA and 0
Principle of Moments: Sum the moments from the components instead b O d a F MO = F d and the direction is counterclockwise, but finding d could be tough F y F MO = (FY a) (FX b), because a and b are easier to find (given) b O a F x The typical sign convention for a moment in 2-D is that counter-clockwise is considered positive.
Moment Calculation: Vector Analysis
Moment Calculation: Vector Analysis While finding the moment of a force in 2-D is straightforward when you know the perpendicular distance d, finding the perpendicular distances can be hard especially when you are working with forces in three dimensions.
Moment Calculation: Vector Analysis While finding the moment of a force in 2-D is straightforward when you know the perpendicular distance d, finding the perpendicular distances can be hard especially when you are working with forces in three dimensions. So a more general approach to finding the moment of a force exists. This more general approach is usually used when dealing with three dimensional forces but can also be used in the two dimensional case as well.
Moment Calculation: Vector Analysis While finding the moment of a force in 2-D is straightforward when you know the perpendicular distance d, finding the perpendicular distances can be hard especially when you are working with forces in three dimensions. So a more general approach to finding the moment of a force exists. This more general approach is usually used when dealing with three dimensional forces but can also be used in the two dimensional case as well. This more general method of finding the moment of a force involves a vector operation called the cross product of two vectors.
Cross Product ( 4.2) In general, the cross product of two vectors A and B results in another vector, C, i.e., C = A X B. The magnitude and direction of the resulting vector can be written as C = A X B = A B sin uc As shown, uc is the unit vector perpendicular to both A and B vectors (or to the plane containing the A and B vectors).
Cross Product The right-hand rule is a useful tool for determining the direction of the vector resulting from a cross product. Calculating dot products between unit vectors is key in BME201: i X j = k j X k = i k X i = j Note that a vector crossed into itself is zero, e.g., i X i = 0
Determinants Also, the cross product can be written as a determinant. Each component can be determined using 2! 2 determinants.
Determinants Also, the cross product can be written as a determinant. Each component can be determined using 2! 2 determinants.
Vector Moment Calculation ( 4.3) Calculate moment about o using MO = r X F where r is the position vector from point O to any point on the line of action of F
r is vector to any point ~ on LOA Find the moment of the weight about o...
2D Example Find the moment of the force about O
2D Example y Find the moment of the force about O x
2D Example y Find the moment of the force about O x
2D Example y Fx Fy Find the moment of the force about O x
2D Example y Fx Fy Find the moment of the force about O Fy = 100 (3/5) N x Fx = 100 (4/5) N
2D Example y Fx Fy Find the moment of the force about O Fy = 100 (3/5) N x Fx = 100 (4/5) N + MO = { 100 (3/5)N (5 m) (100)(4/5)N (2 m)} N m = 460 N m
Principle of Transmissibility A force may be placed anywhere on its line of action without altering the mechanical analysis for a rigid body
Principle of Transmissibility A force may be placed anywhere on its line of action without altering the mechanical analysis for a rigid body
Principle of Transmissibility A force may be placed anywhere on its line of action without altering the mechanical analysis for a rigid body =
Principle of Transmissibility A force may be placed anywhere on its line of action without altering the mechanical analysis for a rigid body = =
Principle of Transmissibility A force may be placed anywhere on its line of action without altering the mechanical analysis for a rigid body y point o y = = point o point o y x x x
Apply the Principles to Reduce Work! F L1 A L2 B
Scalar Analysis in 3D
Scalar Analysis in 3D Moments in 3-D can be calculated using scalar (2-D) approach but it can be difficult and time consuming Must consider every component s moment about every axis and get signs correct.
3D Example Find the moment of F about point O Vector analysis: rbo = {0 i + 3 j + 1.5 k} m MO = rbo X F {0 i + 3 j + 1.5 k} X {-6 i + 3 j + 10 k} To the board... = {25.5 i - 9 j + 18 k} N m
3D Example Find the moment of F about point O Scalar analysis MO = {3(10) 1.5(3)} i 1.5( 6) j + 3(6) k MO = = {25.5 i - 9 j + 18 k} N m
3D Example Find the moment of F about point O Scalar analysis MO = {3(10) 1.5(3)} i 1.5( 6) j + 3(6) k RHR MO = = {25.5 i - 9 j + 18 k} N m
Attention Quiz If M = r X F, then what will be the value of M r? A) 0!! B) 1 C) r 2 F! D) None of the above
Attention Quiz If M = r X F, then what will be the value of M r? A) 0!! B) 1 C) r 2 F! D) None of the above 10 N 3 m P 2 m 5 N Using the CCW direction as positive, the net moment of the two forces about point P is A) 10 N!m B) 20 N!m C) - 20 N!m D) 40 N!m E) - 40 N!m
Example 1
Example 1
Example 2
Example 2
Example 2
Example 2 To the board..
Example 4.26
4.43 Find the moment produced by F A about point O located on the drill bit