Physics 170 Lecture 9. We all have our moments...

Transcription

1 Phys 170 Lecture 9 1 Physics 170 Lecture 9 Chapter 4 - Force System Resultants We all have our moments...

2 Moment of a Force in 2D M = ±RF sinθ = ±RF = ±Fd = R x F y R y F x Use which ever is easiest, they all give the same value. Treat F, R, d, and θ as positive. F Use + sign if rotation would be + (CCW), use sign if rotation would be (CW). The last form calculates the sign. Pivot Point Moment Arm Application Point Phys 170 Lecture 9 2 y θ d R x F Force Line θ

3 Sliding Along the Force Line Doesn t Change the Moment The moment arm d is the same if we slide the force along its own force line (with constant magnitude and direction). So the moment is the same if we slide the force along its line. Pivot Point Force Line Phys 170 Lecture 9 34 d F F F Application Points

4 Phys 170 Lecture 9 4 Force Couple and Couple Moment Forces that are equal and opposite, but with different application points are called a force couple. F 1 The forces cancel, but they still produce d d what is called a couple moment. If the distance between the force lines is, the sum of the moments F 2 = d F 1 is M 1 + M 2 = F d for any choice of pivot point. You need to put in the sign by hand (this one is positive). The book uses a blue curved arrow to symbolize a couple moment.

5 Moment Vector in 3D We have an R vector, and an F vector, and they define a plane. In that plane, use the 2D definition of the moment magnitude M = RF = RF sinθ The moment vector is perpendicular to the plane defined by R and F. It points in the direction of your thumb using the right-hand-rule from R to F. Phys 170 Lecture 9 F R 5

6 Its z-component is identical to the 2-D cross product we invented for doing 2-D moment calculations. Phys 170 Lecture 9 3-D Cross Product The 3-D cross product of two vectors is written C = A B. C x = A y B z A z It has components C y = A z B x A x B z It is perpendicular to both C z = A x A y B x A and B (by construction) It is proportional to the product of the magnitudes of (at least this symmetrized version is). A and B

7 3-D Cross Product (2) If A and B are identical, A B = 0. C x = A y B z A z If A and B are parallel, then A = k C y = A z B x A x B z B, and we still have A B C z = A x A y B x = 0. If A and B are opposite directions, then A = kb with k < 0, and it s still true that A B = 0. With a bit of algebra, we can show But flipping the order flips the sign: If A and B are in the x-y plane, then A z = B z = 0, and the x and y î and ĵ components are both zero. Phys 170 Lecture 9 ( ) A B + C ( ) = A B + A C A B = B A

8 3-D Cross Product (3) We can calculate the 3-D cross product with a determinant trick î ĵ ˆk A x A y A z = î A yb z A z B x B z ( ) + ĵ A zb x A x B z Wrap around, down to right is + sign, down to left is sign. The each component of C is the product of the other two components of A and B, and not the same component. There is a cyclic order x y z x y and the sign flips if the order flips. Phys 170 Lecture 9 ( ) ( ) + ˆk A x A y B x

9 It s not much faster than manual, but more reliable. Phys 170 Lecture 9 3-D Cross Product on Calculators Many calculators know about vectors, and can do the crossproduct of two vectors (also dot-product, add, subtract...) Casio fx-991ms, Casio fx-991es, Sharp EL-W516X, TI-83, TI-84, TI-89, TI nspire can all do vectors (but you have to write a program for cross- and dot-product for the TI-83 or TI-84). There is a link on the course web page to a text file that gives instructions on doing vector operations on calculators, including the programs needed on TI-83 and TI-84.

10 Moment Around an Axis in 3D We define an axis in 3D by a unit vector û The moment around that axis is just the dot-product of the 3D moment vector and the unit vector: M axis = û M = û ( R F ) Note that the tail-end of the R vector must be on the axis. We can calculate the components of M by the usual determinant rule, then do the dot-product: M axis = û x M x + û y M y + û z M z Phys 170 Lecture 9 10 û r F

11 Phys 170 Lecture 9 11 Determine the moment of F about A. Determine the lever arm for F. Determine the magnitude of the moment of F about an axis extending from A to C.

12 Determine the moment of F about A. R = 4, 3, 2 ft ( ) Determine the lever arm for F. F = ( 4, 12, 3) lb Determine the magnitude of the moment of F about an axis M = R F extending from A to C. ˆ i ( ) ( ) (page 146, 13th edition) ˆ PROBLEM i j k 4-53 M = = ( 2 ) 4 4 ( 3) j + th PROBLEM 4-55 (page 145, 12 edition) [ ] k Determine the) moment of F about A. = (15, 4, 36 ft-lb Determine the lever arm for F M moment + 36 of F39.20 Determine the magnitude of the about an axis d= = = = ft 2 13 extending from AFto C ( 3) Phys 170 Lecture 9 12

13 Phys 170 Lecture 9 13 Determine the magnitude of the moment of F about an axis extending from A to C. Dot-product of moment vector with unit vector along AC. A = ( 0, 0, 0) C = ( 4, 3, 0) û = ( 4, 3, 0 ) ( ) = 4, 3, M = 15, 4, 36 ( ) û M = = 72 5 = 14.4 ft-lb