Phys 170 Lecture 4 1. Physics 170 Lecture 4. Solving for 3 Force Magnitudes in Known Directions

Transcription

1 Phys 170 Lecture 4 1 Physics 170 Lecture 4 Solving for 3 Force Magnitudes in Known Directions

2 Phys 170 Lecture 4 2 Mastering Engineering Issues There is definitely a problem with the campusebookstore.com company that is the middle-man between UBC Bookstore and Pearson. Some people get their access code instantly from the web. Others are told to wait for an , which never comes. We are going to move the assignment due dates back.

3 Phys 170 Lecture 4 3

4 Force Along Line Between Points 1. Position difference vector r = (end point) (start point) r = ( x end x start )î + ( y y ) ĵ + ( z z ) ˆk end start end start 2. Length of position difference vector Lr L r = r x 2 + r y 2 + r z 2 = ( x end x ) 2 start + ( y end y ) 2 start + ( z end z ) 2 start 3. Unit vector û r = r xî + r y ĵ + r z ˆk L r 4. Force vector = (force-magnitude) times unit-vector F = Fû r Phys 170 Lecture 4 4

5 Phys 170 Lecture 4 5 Setting Up Equations F R = + + F R = û B + û C + û D = u Bx + u Cx + u Dx = u By + u Cy + u Dy = u Bz + u Cz + u Dx = u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz

6 Phys 170 Lecture 4 6 Setting Up Equations F R = + + F R = û B + û C + û D = u Bx + u Cx + u Dx = u By + u Cy + u Dy = u Bz + u Cz + u Dx = u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz

7 Phys 170 Lecture 4 7 Setting Up Equations F R = + + F R = û B + û C + û D = u Bx + u Cx + u Dx = u By + u Cy + u Dy = u Bz + u Cz + u Dx = u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz

8 Phys 170 Lecture 4 8 Solve by Gaussian Elimination = u Bx = u By + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

9 Phys 170 Lecture 4 9 Solve by Gaussian Elimination = u Bx = u By + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

10 Phys 170 Lecture 4 10 Solve by Gaussian Elimination = 1 = u By + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

11 Phys 170 Lecture 4 11 Solve by Gaussian Elimination = 1 = u By + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

12 Phys 170 Lecture 4 12 Solve by Gaussian Elimination = 1 = 0 + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

13 Phys 170 Lecture 4 13 Solve by Gaussian Elimination = 1 = 0 + u Cy = u Bz + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

14 Phys 170 Lecture 4 14 Solve by Gaussian Elimination = 1 = 0 + u Cy = 0 + u Cz Divide x-eqn by ubx, so upper left term is just 1 times FB Multiply new x eqn by uby and subtract that from y-eqn, so first term of new y-eqn is zero. Multiply new x eqn by ubz and subtract that from z-eqn, so first term of new z-eqn is zero.

15 Phys 170 Lecture 4 15 Solve by Gaussian Elimination = 1 = 0 + u Cy = 0 + u Cz Divide y-eqn by u Cx, so upper middle term is just 1 times FC Multiply new y eqn by u Cz and subtract that from z-eqn, so first 2 terms of new y-eqn are zero.

16 Phys 170 Lecture 4 16 Solve by Gaussian Elimination = 1 = 0 +1 = 0 + u Cz Divide y-eqn by u Cx, so upper middle term is just 1 times FC Multiply new y eqn by u Cz and subtract that from z-eqn, so first 2 terms of new y-eqn are zero.

17 Phys 170 Lecture 4 17 Solve by Gaussian Elimination = 1 = 0 +1 = 0 + u Cz Divide y-eqn by u Cx, so upper middle term is just 1 times FC Multiply new y eqn by u Cz and subtract that from z-eqn, so now first two terms of new z-eqn are zero.

18 Phys 170 Lecture 4 18 Solve by Gaussian Elimination = 1 = 0 +1 = u Dz The z-equation is easy to solve for FD. Plug FD into y-equation and solve for FC. Plug FD and FC into x-equation and solve for FB.

19 Phys 170 Lecture 4 19 Diagonalization = 1 = 0 +1 = u Dz Divide the last equation by u Dz to make bottom right coef = 1.

20 Phys 170 Lecture 4 20 Diagonalization = 1 = 0 +1 = Divide the last equation by u Dz to make bottom right coef = 1.

21 Phys 170 Lecture 4 21 Diagonalization = 1 = 0 +1 = Divide the last equation by u Dz to make bottom right coef = 1. Eliminate the coefficients above the diagonal in the obvious way.

22 Phys 170 Lecture 4 22 Diagonalization = = = Divide the last equation by u Dz to make bottom right coef = 1. Eliminate the coefficients above the diagonal in the obvious way.

23 Phys 170 Lecture 4 23 Matrix Equivalent = u Bx = u By + u Cy = u Bz + u Cz = u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz

24 Phys 170 Lecture 4 24 The Augmented Matrix = u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz

25 Phys 170 Lecture 4 25 Reduced Row Echelon Form = u Bx = u By + u Cy = u Bz + u Cz u Bx u Cx u Dx u By u Cy u Dy u Bz u Cz u Dz U-matrix RREF

26 Phys 170 Lecture 4 26 Calculator Linear Equation Instructions Linear Equations on calculators Casio fx-991ms This calculator can solve linear equations for 2 or 3 variables. Enter "equation mode" by pressing MODE MODE MODE then 1 (EQN). A small EQN at the top of the display indicates that you are in "equation mode" Going into "calc mode" or "vector mode" or "matrix mode" erases any equation information. Normal calculations are not possible in "equation mode," except to calculate equation coefficients. After entering "equation mode", the screen shows Unknowns? -> (right-cursor for quadratic or cubic equations) 2 3 Enter 3 for 3 unknowns, and the screen shows a1? 0. Enter the first coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows b1? 0. Enter the second coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows c1? 0. Enter the third coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows d1? 0. Enter the right-hand-side constant for the first equation (or a calculation for it) and =(equals), and the screen shows a2? 0. Continue this process of entering coefficients and right-hand-sides for the second equation, and third equation. After the last value (right hand side of third equation) and =(equals), the screen shows x= xx.xxxxxxxxxxx the solution for the first variable. Hit =(equals) and the screen shows y= yy.yyyyyyyyyyy the solution for the second variable. Hit =(equals) and the screen shows z= zz.zzzzzzzzzzz the solution for the third variable. Hit =(equals) and the screen shows a1= aa.aa1 the first coefficient that you entered You can use the up and down cursors to scroll through the coefficients and right-hand-sides, to check or edit them. You can also use the cursors to scroll through the solutions. The AC (all-clear) button does not clear the coefficients, nor does turning the calculator off. Exiting "equation mode" and re-entering does clear them (sets them all to zero).

27 R-matrix Using r s instead of u s F R = rb L B + r C + L C Phys 170 Lecture 4 27 r D L D = r Bx L B + r Cx L C + r Dx L D = r By L B + r Cy L C + r Dy L D = r Bz L B + r Cz L C + r Dx L D = r Bx r Cx r Dx r By r Cy r Dy r Bz r Cz r Dz L B L C L D

28 Phys 170 Lecture 4 28 Using r s instead of u s First step is finding the position difference vectors r 1, r 2, r 2 Next step is finding their lengths L1, L2, L3 But instead of doing 9 divisions to go from components of r 1, r 2, r 2 to components of û 1, û 2, û 2 to make the U-matrix which will solve for the F s directly, use the raw r1, r 2, r 2 components directly to make the R-matrix. Applying RREF to the augmented R-matrix gives so we need to do 3 multiply s at the end to get the forces. F 1 L 1, F 2 L 2, F 3 L 3