ME 201 Engineering Mechanics: Statics. Unit 1.2 Scalars and Vectors Vector Operations Vector Addition of Forces

Transcription

1 ME 201 Engineering Mechanics: Statics Unit 1.2 Scalars and Vectors Vector Operations Vector ddition of Forces

2 dditional ssistance Tutoring Center Mck 272 Engineering Mechanics Help Lab us 9

3 Scalar Scalars & Vectors positive or negative number magnitude only Vector magnitude and direction

4 Scalars & Vectors Vectors Quantity with both magnitude & direction Does NOT follow elementary arithmetic/algebra rules Examples Head Direction/ngle Tail Line of ction

5 Parallelogram Law The resultant of two forces can be obtained by Joining the vectors at their tails Constructing a parallelogram R The resultant is the diagonal of the parallelogram

6 Triangle Construction The resultant of two forces can be obtained by Joining the vectors in tip-to-tail fashion The resultant extends R from the tail of to the head of the

7 Solving Problems Use parallelogram law or triangle construction to construct force triangle Use trigonometry to find unknowns: Law of sines sin a a C sin b b c C sin c Law of cosines C cos( c)

8 Example Problem Triangle Law Given: F 1 = 100 N F 2 = 150 N 10º F 2 Find: R F 1 15º

9 Example Problem Solution Triangle Law 10º F 2 F 1 15º Solution: C cos( c) R F 1 θ 15º 10º F 2 R cos(115 ) R N How would you find the angle R makes with the x-axis?

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12 Review Exercise missionary purchased 4.5 kg of bananas. How many pounds did he purchase? Round your answer to 3 significant digits.

13 Video 1d Fundamental Concepts Scalars numeric value Vectors magnitude & direction

14 Video 1e Fundamental Concepts Vector Operations Multiplication of a Vector by a scalar Vector ddition Parallelogram Law R Vector ddition Triangle Construction R

15 Vector Operations Given vectors and as shown below, which figure best shows 2-. (D) (E) () () (C)

16 Vector Operations Given vectors and as shown below, which figure best shows (D) (E) () () (C)

17 Video 1f Fundamental Concepts Properties of Vectors Vector ddition IS Commutative + = + Vector Subtraction IS NOT Commutative - - Solving Problems Construct Parallelogram/Triangle Law of Sines Law of Cosines sin a C sin b 2 2 C sin c 2 cos( c)

18 Key Concepts: Force Vector asics Scalar vs. Vector Vector Operations Problem Solving pproaches Parallelogram Law & Triangle Construction

19 Scalar vs. Vector What is the difference between a scalar and a vector?. Scalars and Vectors both have a magnitude. Scalars and Vectors both have a direction C. Only Scalars have Direction D. Only Vectors have Direction E. oth and D

20 Scalar Scalars & Vectors positive or negative number magnitude only Vector magnitude and direction

21 Scalars & Vectors Vectors Quantity with both magnitude & direction Does NOT follow elementary arithmetic/algebra rules Examples Head Direction/ngle Tail Line of ction

22 Scalar vs. Vector Key Concepts Identify which are Scalar and which are Vector quantities: Volume Velocity Mass Length Force Moment Scalar Vector Scalar Scalar Vector Vector

23 Vector Operations Key Concepts Can we multiply a vector by a scalar?

24 Vector Operations Multiplication & Division of Vector () by Scalar (a) a * = a 2 * = * =

25 Vector Operations Given vectors and as shown below, which figure best shows 2-. (D) (E) () () (C)

26 Vector Operations Given vectors and as shown below, which figure best shows (D) (E) () () (C)

27 Vector Operations Key Concepts How do we add vectors? Parallelogram Law Triangle Construction Cartesian Vectors

28 Parallelogram Law The resultant of two forces can be obtained by Joining the vectors at their tails Constructing a parallelogram R The resultant is the diagonal of the parallelogram

29 Triangle Construction The resultant of two forces can be obtained by Joining the vectors in tip-to-tail fashion The resultant extends R from the tail of to the head of the

30 Solving Problems Use parallelogram law or triangle construction to construct force triangle Use trigonometry to find unknowns: Law of sines sin a a C sin b b c C sin c Law of cosines C cos( c)

31 Vector ddition Does + = +? R R YES! - commutative

32 Vector Subtraction Does = -? - R -R - NO! opposite sense

33 Vector Subtraction - = + (-) - R -

34 Example Problems

35 Example Problem Triangle Law Given: F 1 = 100 N F 2 = 150 N 10º F 2 Find: R F 1 15º

36 Example Problem Solution 10º F 2 F 1 15º

37 Example Problem Solution Triangle Law 10º F 2 F 1 15º Solution: C cos( c) R F 1 θ 15º 10º F 2 R cos(115 ) R N How would you find the angle R makes with the x-axis?

38 Graphical Solutions Key Concepts When should graphical solutions be used?

39 In Class Exercise

40 Solution: 2-15

41 In Class Exercise

42 Solution: 2-27