Force Vector (cont’)

2.8 Force Vector Directed along a Line

Force F acting along the chain can be presented as a Cartesian vector by- Establish x, y, z axes- Form a position vector r along length of chain

Unit vector, u = r/r that defines the direction of both the chain and the force

We get F = Fu

Example 2.13

The man pulls on the cord with a force of 350N. Represent this force acting on the support A, as a Cartesian vector and determine its direction.

SolutionEnd points of the cord are A (0m, 0m, 7.5m) and B (3m, -2m, 1.5m)r = (3m – 0m)i + (-2m – 0m)j + (1.5m – 7.5m)k = {3i – 2j – 6k}m

Magnitude = length of cord ABUnit vector, u = r /r = 3/7i - 2/7j - 6/7k

mmmmr 7623 222

Solution

Force F has a magnitude of 350N, direction specified by u.F = Fu = 350N(3/7i - 2/7j - 6/7k) = {150i - 100j - 300k} N

α = cos-1(3/7) = 64.6°

β = cos-1(-2/7) = 107°

γ = cos-1(-6/7) = 149°

2.9 Dot Product

Dot product of vectors A and B is written as A·B (Read A dot B)

Define the magnitudes of A and B and the angle between their tails

A·B = AB cosθ where 0°≤ θ ≤180°

Referred to as scalar product of vectors as result is a scalar

2.9 Dot Product

Laws of Operation1. Commutative law

A·B = B·A2. Multiplication by a scalar

a(A·B) = (aA)·B = A·(aB) = (A·B)a3. Distribution law

A·(B + D) = (A·B) + (A·D)

2.9 Dot Product

Cartesian Vector Formulation- Dot product of Cartesian unit vectors

i·i = (1)(1)cos0° = 1i·j = (1)(1)cos90° = 0

- Similarlyi·i = 1 j·j = 1 k·k = 1 i·j = 0 i·k = 1 j·k = 1

2.9 Dot ProductCartesian Vector Formulation

Dot product of 2 vectors A and B

A·B = AxBx + AyBy + AzBz

Applications The angle formed between two vectors or

intersecting lines.θ = cos-1 [(A·B)/(AB)] 0°≤ θ ≤180°

The components of a vector parallel and perpendicular to a line.

Aa = A cos θ = A·u

Example 2.17

The frame is subjected to a horizontal force F = {300j} N. Determine the components of this force parallel and perpendicular to the member AB.

Solution

Since

Thus

N

kjijuF

FF

kji

kjirr

u

B

AB

B

BB

1.257

)429.0)(0()857.0)(300()286.0)(0(

429.0857.0286.0300.

cos

429.0857.0286.0

362

362222

Solution

Since result is a positive scalar, FAB has the same sense of direction as uB. Express in Cartesian form

Perpendicular component

NkjikjijFFF

Nkji

kjiN

uFF

AB

ABABAB

}110805.73{)1102205.73(300

}1102205.73{

429.0857.0286.01.257

Solution

Magnitude can be determined from F┴

or from Pythagorean Theorem,

N

NN

FFF AB

155

1.257300 22

22

Quiz

1. Which one of the following is a scalar quantity? A) Force B) Position C) Mass D) Velocity

2. For vector addition, you have to use ______ law. A) Newton’s Second B) the arithmetic C) Pascal’s D) the

parallelogram

3. Can you resolve a 2-D vector along two directions, which are not at 90° to each other?

A) Yes, but not uniquely. B) No. C) Yes, uniquely.

4. Can you resolve a 2-D vector along three directions (say at 0, 60, and 120°)?

A) Yes, but not uniquely. B) No. C) Yes, uniquely.

5. Resolve F along x and y axes and write it in vector form. A) 80 cos (30°) i – 80 sin (30°) j B) 80 sin (30°) i + 80 cos (30°) j C) 80 sin (30°) i – 80 cos (30°) j D) 80 cos (30°) i + 80 sin (30°) j

30°

xy

F = 80 N

6. Determine the magnitude of the resultant (F1 + F2) force in N when F1={ 10i + 20j }N and F2={ 20i + 20j } N .

A) 30 N B) 40 N C) 50 N D) 60 N E) 70 N

7. Vector algebra, as we are going to use it, is based on a ___________ coordinate system.

A) Euclidean B) Left-handed C) Greek D) Right-handed E) Egyptian

8. The symbols , , and designate the __ of a 3-D Cartesian vector.

A) Unit vectors B) Coordinate direction angles C) Greek societies D) X, Y and Z components

9. What is not true about an unit vector, uA ? A) It is dimensionless. B) Its magnitude is one. C) It always points in the direction of positive X- axis. D) It always points in the direction of vector A.

10. If F = {10 i + 10 j + 10 k} N and G = {20 i + 20 j + 20 k } N, then F + G = { ____ } N

A) 10 i + 10 j + 10 k B) 30 i + 20 j + 30 k C) – 10 i – 10 j – 10 k D) 30 i + 30 j + 30 k

12. A force of magnitude F, directed along a unit vector U, is given by F =

A) F (U) B) U / F C) F / U D) F + U (E) F – U

13. P and Q are two points in a 3-D space. How are the position vectors rPQ and rQP related?

A) rPQ = rQP B) rPQ = - rQPC) rPQ = 1/rQP D) rPQ = 2 rQP

14. If F and r are force vector and position vectors, respectively, in SI units, what are the units of the expression (r * (F / F)) ?

A) Newton B) Dimensionless C) Meter D) Newton – Meter

E) The expression is algebraically illegal.

15. Two points in 3 – D space have coordinates of P (1, 2, 3) and Q (4, 5, 6) meters. The position vector rQP is given by

A) {3 i + 3 j + 3 k} m B) {– 3 i – 3 j – 3 k} mC) {5 i + 7 j + 9 k} m D) {– 3 i + 3 j + 3 k} mE) {4 i + 5 j + 6 k} m

16. Force vector, F, directed along a line PQ is given byA) (F/ F) rPQ B) rPQ/rPQ C) F(rPQ/rPQ) D) F(rPQ/rPQ)

17. The dot product of two vectors P and Q is defined as A) P Q cos B) P Q sin C) P Q tan D) P Q sec

18. The dot product of two vectors results in a _________ quantity. A) Scalar B) Vector C) Complex D) Zero

19. If a dot product of two non-zero vectors is 0, then the two vectors must be _____________ to each other.

A) Parallel (pointing in the same direction)B) Parallel (pointing in the opposite direction)C) PerpendicularD) Cannot be determined.

20. If a dot product of two non-zero vectors equals -1, then the vectors must be ________ to each other.

A) Parallel (pointing in the same direction)B) Parallel (pointing in the opposite direction)C) PerpendicularD) Cannot be determined.

21. The dot product can be used to find all of the following except ____ .

A) sum of two vectorsB) angle between two vectorsC) component of a vector parallel to another lineD) component of a vector perpendicular to another line

22. Find the dot product of the two vectors P and Q. P = {5 i + 2 j + 3 k} m Q = {-2 i + 5 j + 4 k} m

A) -12 m B) 12 m C) 12 m2 D) -12 m2 E) 10 m2

Next Chapter...

Equilibrium of a particle

Chapter Objectives

Concept of the free-body diagram for a particle

Solve particle equilibrium problems using the equations of equilibrium

Chapter Outline

1. Condition for the Equilibrium of a Particle

2. The Free-Body Diagram3. Coplanar Systems4. Three-Dimensional Force Systems

3.1 Condition for the Equilibrium of a Particle

Particle at equilibrium if- At rest- Moving at constant a constant velocity

Newton’s first law of motion∑F = 0

where ∑F is the vector sum of all the forces acting on the particle

3.1 Condition for the Equilibrium of a Particle

Newton’s second law of motion∑F = ma

When the force fulfill Newton's first law of motion,

ma = 0 a = 0

therefore, the particle is moving in constant velocity or at rest

3.2 The Free-Body Diagram

Best representation of all the unknown forces (∑F) which acts on a body

A sketch showing the particle “free” from the surroundings with all the forces acting on it

Consider two common connections in this subject – Spring Cables and Pulleys

3.2 The Free-Body Diagram

Spring Linear elastic spring: change in length is

directly proportional to the force acting on it spring constant or stiffness k: defines the

elasticity of the spring Magnitude of force when spring

is elongated or compressed

F = ks

3.2 The Free-Body Diagram

Cables and Pulley Cables (or cords) are assumed negligible

weight and cannot stretch Tension always acts in the direction of the

cable Tension force must have a constant

magnitude for equilibrium For any angle θ, the cable

is subjected to a constant tension T

Procedure for Drawing a FBD

1. Draw outlined shape

2. Show all the forces- Active forces: particle in motion- Reactive forces: constraints that prevent motion

3. Identify each forces- Known forces with proper magnitude and direction- Letters used to represent magnitude and directions

Example 3.1

The sphere has a mass of 6kg and is supported. Draw a free-body diagram of the sphere, the cord CE and the knot at C.

Solution

FBD at SphereTwo forces acting, weight and the force on cord CE. Weight of 6kg (9.81m/s2) = 58.9NCord CETwo forces acting: sphere and knotNewton’s 3rd Law: FCE is equal but opposite

FCE and FEC pull the cord in tension

For equilibrium, FCE = FEC

Solution

FBD at Knot3 forces acting: cord CBA, cord CE and spring CD Important to know that the weight of the sphere does not act directly on the knot but subjected to by the cord CE

3.3 Coplanar SystemsA particle is subjected to coplanar forces

in the x-y planeResolve into i and j components for

equilibrium ∑Fx = 0

∑Fy = 0Scalar equations of equilibrium

require that the algebraic sum of the x and y components to equal to zero

Procedure for Analysis

1. Free-Body Diagram- Establish the x, y axes- Label all the unknown and known forces

2. Equations of Equilibrium- Apply F = ks to find spring force - When negative result force- indicates its sense is reverse of that shown on the free body diagram- Apply the equations of equilibrium

∑Fx = 0 ∑Fy = 0

Example 3.4

Determine the required length of the cord AC so that the 8kg lamp is suspended. The undeformed length of the spring AB is l’AB = 0.4m, and the spring has a stiffness of kAB = 300N/m.

Solution

1. Draw FBD at Point AThree forces acting, force by cable AC, force in spring AB and weight of the lamp.If force on cable AB is known, stretch of the spring is found by F = ks. +→ ∑Fx = 0; TAB – TAC cos30º = 0

+↑ ∑Fy = 0; TABsin30º – 78.5N = 0

Solving, TAC = 157.0kN

TAB = 136.0kN

Solution

TAB = kABsAB; 136.0N = 300N/m(sAB)

sAB = 0.453N

For stretched length, lAB = l’AB+ sAB

lAB = 0.4m + 0.453m

= 0.853mFor horizontal distance BC, 2m = lACcos30° + 0.853m

lAC = 1.32m

3.4 Three-Dimensional Force SystemsFor particle equilibrium

∑F = 0Resolving into i, j, k components

∑Fxi + ∑Fyj + ∑Fzk = 0Three scalar equations representing

algebraic sums of the x, y, z forces ∑Fxi = 0

∑Fyj = 0

∑Fzk = 0

Procedure for Analysis Free-body Diagram

- Establish the z, y, z axes - Label all known and unknown force

Equations of Equilibrium- Apply ∑Fx = 0, ∑Fy = 0 and ∑Fz = 0

- Substitute vectors into ∑F = 0 and set i, j, k components = 0

- Negative results indicate that the sense of the force is opposite to that shown in the FBD.

Example 3.7

Determine the force developed in each cable used to support the 40kN crate.

Solution1. Draw FBD at Point ATo expose all three unknown forces in the cables.2. Equations of EquilibriumExpressing each forces in Cartesian vectors, FB = FB(rB / rB)

= -0.318FBi – 0.424FBj + 0.848FBk FC = FC (rC / rC)

= -0.318FCi – 0.424FCj + 0.848FCk FD = FDi

W = -40k

])8()4()3(

843[

222

kjiBF

SolutionFor equilibrium, ∑F = 0; FB + FC + FD + W = 0

-0.318FBi – 0.424FBj + 0.848FBk - 0.318FCi

– 0.424FCj + 0.848FCk + FDi - 40k = 0

∑Fx = 0; -0.318FB - 0.318FC + FD = 0

∑Fy = 0; – 0.424FB – 0.424FC = 0

∑Fz = 0; 0.848FB + 0.848FC - 40 = 0

Solving, FB = FC = 23.6kN

FD = 15.0kN

QUIZ

1. When a particle is in equilibrium, the sum of forces acting on it equals ___ . (Choose the most appropriate answer)

A) A constant B) A positive number C) Zero D) A negative number E) An integer

2. For a frictionless pulley and cable, tensions in the cables are related as

A) T1 > T2

B) T1 = T2

C) T1 < T2

D) T1 = T2 sin T1

T2

3. Assuming you know the geometry of the ropes, you cannot determine the forces in the cables in which system above?

4. Why?A) The weight is too heavy.B) The cables are too thin.C) There are more unknowns than equations.D) There are too few cables for a 100 kg

weight.

100 N100 N 100 N

( A ) ( B ) ( C )

5. Select the correct FBD of particle A.

A 40

100 kg

30

30A) A

100 kg

B)40°

A

F1 F2

C) 30°A

F

100 kg

A

30° 40°F1 F2

100 kg

D)

6. Using this FBD of Point C, the sum of forces in the x-direction ( FX) is ___ . Use a sign convention of + .

A) F2 sin 50° – 20 = 0

B) F2 cos 50° – 20 = 0

C) F2 sin 50° – F1 = 0

D) F2 cos 50° + 20 = 0

50o

C

F2

20kN

7. In 3-D, when a particle is in equilibrium, which of the following equations apply?A) ( Fx) i + ( Fy) j + ( Fz) k = 0

B) F = 0

C) Fx = Fy = Fz = 0

D) All of the above.

E) None of the above.

8. In 3-D, when you know the direction of a force but not its magnitude, how many unknowns corresponding to that force remain?

A) One B) Two C) Three D) Four

9. In 3-D, when you don’t know the direction or the magnitude of a force, how many unknowns do you have corresponding to that force?

A) One B) Two C) Three D) Four

10. Four forces act at point A and point A is in equilibrium. Select the correct force vector P.

A) {-20 i + 10 j – 10 k}NB) {-10 i – 20 j – 10 k} NC) {+ 20 i – 10 j – 10 k}ND) None of the above.

z

F3 = 10 NP

x

A

F2 = 10 N

y

F1 = 20 N

Answer

1.C2.B3.C4.C5.D6.B7.D8.A9.C10.D