Concurrent Vector Systems in Cartesian Coordinates

Transcription

1 Concurrent Vector Systems in Cartesian Coordinates CARPERPETUATION (kar' pur pet u a shun) n. The act, when vacuuming, of running over a string or a piece of lint at least a dozen times, reaching over and picking it up, examining it, then putting it back down to give the vacuum one more chance. Review Two forces can be combined using the parallelogram law to form a resultant A resultant can be broken up into its components using the geometry of the system and some trig 2 1

2 Determining Components Lets look at the components of F 1 3 Determining Components Last class we drew the angle, used the trigonometric relaaonships to determine the magnitudes of the components, and then used the direcaons from the drawing to determine the signs of the components 4 2

3 Determining Components Rather than using β for our calculaaons, we could look at the angle formed CCW with the +x- axis We will label this angle θ 5 Determining Components With the angle measured this way, the following will always hold true if the angle is measured CCW from the +x axis 6 3

4 Determining Components NoAce the absolute value signs We are only using the magnitude of F 1 7 Determining Components In this case, when you perform the calculaaon, you will get a sign on the result for the component 8 4

5 Determining Components Therefore, you will not have to take the direcaon from the drawing, it will come from the calculaaon 9 Determining Components Don t use both methods, that can lead to mistakes in the direcaons of the components 10 5

6 Determining Components You can also use this type of logic, to determine the direcaon of the resultant of a series of components 11 Determining Components You can also use this type of logic, to determine the direcaon of the resultant of a series of components 12 6

7 Determining Components If we take the second equaaon and divide it by the first equaaon, the magnitudes of the resultant drop out 13 Determining Components Two things to remember here You must use the signs of the components to get the correct angle 14 7

8 Determining Components Two things to remember here The tangent func4on on your calculator may not be very smart 15 Determining Components If you don t pay apenaon, your calculator may give you the same result for both components as + as it will for both components as

9 Determining Components If you don t pay apenaon, your calculator may give you the same result if each of the components is of a different sign. 17 Determining Components You are smarter than your calculator, think about the sensibility of your soluaon. 18 9

10 Resultant of a Series of Vectors The really nice part of this comes when we take a series of forces at a point and develop a single resultant from all the forces 19 Resultant of a Series of Vectors We will start with three forces F 1, F 2, and F 3 and try and find the resultant, F which is the vector sum of the three forces 20 10

11 Resultant of a Series of Vectors We have already moved (or the forces were already posiaoned) so that their tails were at the same point 21 Resultant of a Series of Vectors The resultant F is the vector sum of the three forces, F 1, F 2, and F

12 Resultant of a Series of Vectors If we start by forming the resultant of F 1 and F 2 which we will call F 4 23 Resultant of a Series of Vectors We can use informaaon that (hopefully) is given in either the iniaal informaaon or can be deduced from the geometry of the system to form components of F 1 and F

13 Resultant of a Series of Vectors We can use informaaon that (hopefully) is given in either the iniaal informaaon or can be deduced from the geometry of the system to form components of F 1 and F 2 25 Resultant of a Series of Vectors Considering the two forces in components

14 Resultant of a Series of Vectors We have the components but we have to be careful here about how we combine them 27 Resultant of a Series of Vectors We have a sign convenaon that takes the direcaon of the force into consideraaon Note that all of these calculaaons result in scalar quanaaes 28 14

15 Resultant of a Series of Vectors Our sign convenaon follows the axis labeling The + direcaon on each axis as going from the origin to the axis label 29 Resultant of a Series of Vectors Since the y is labeled on the axis going up, this is the posiave y direcaon which means any force in that direcaon will be considered posiave 30 15

16 Resultant of a Series of Vectors By the same reasoning, any force directed to the right (toward the x label) is considered as posiave 31 Resultant of a Series of Vectors To find the components of the resultant, we can now ualize our sign convenaon and normal algebraic addiaon (rather than vector addiaon) 32 16

17 Resultant of a Series of Vectors We now have two force vectors F 4 and F 3 which are to be combined into a resultant force 33 Resultant of a Series of Vectors And subsatuang our trigonometric evaluaaons 34 17

18 Resultant of a Series of Vectors If we maintain a consistent sign convenaon, we can make a general statement about finding the resultant of any n vectors whose lines of acaon intersect at a point 35 Resultant of a Series of Vectors 36 18

19 Resultant of a Series of Vectors Remember that this is only true if we maintain a consistent sign convenaon and take the algebraic sign from our axis selecaon 37 Sample Problem 1 F2-8. Determine the magnitude and direcaon of the resultant force

20 Sample Problem If φ = 30 and the resultant force acang on the gusset plate is directed along the posiave x- axis, determine the magnitudes of F 2 and the resultant force. 39 Sample Problem Determine the magnitude of the resultant force and its direcaon measured CCW from the posiave x- axis