Statics and Strength of Materials: Force & Position Vectors Part i

CIV ENG 2P04: Statics and Strength of Materials - Lecture 2: CH2: Force & Position Vectors Part i

Course Overview

Statics (Weeks 1 to 6)

  • CH: 1, 2, 3: Particle Equilibrium, Force Vectors, Position Vectors & Force System Resultants

    • 3D Force vectors

  • CH: 4: Equilibrium of Rigid Bodies

    • Static (in)determinacy

    • Stability

    • Free body diagrams

  • CH: 5: Centre of Gravity; Centroids & Moment of Inertia

  • CH: 6: Structural Analysis

    • Internal Forces

    • Analysis of trusses

Mechanics of Materials (Weeks 7 to 13)

  • CH: 7: Stress & Strain

  • CH: 8: Mechanical Properties of Materials

  • CH: 9: Axial Load

  • CH: 10: Torsion

    • Shear stress & strain

    • Torsion of solid and hollow circular sections

  • CH: 11: Bending

  • CH: 12: Transverse Shear

  • CH: 13: Combined Loading

  • CH: 14: Stress Transformation

    • Principal stresses

    • Mohr's circle representation

Lecture Outline (CH2: Force & Position Vectors Part i)

  • Scalars and Vectors

  • Vector Operations

    • Addition

    • Subtraction

    • Parallelogram Law

    • Trigonometric Approach

  • Vector Addition of Forces

    • Resultant Force

    • Force Component

  • Addition of a System of Coplanar Forces

    • Scalar notation

    • Cartesian Notation

Learning Outcomes

By the end of this chapter, you should be able to:

  • Add forces and resolve them into components.

  • Express force and position as Cartesian vectors.

  • Use the dot product to find the angle between two vectors or the projection of one vector onto another. (Note: Dot product topics are typically covered in subsequent parts of this chapter).

Previous Knowledge & Real-World Relevance

  • Application to Trusses:

    • By the end of this week, we will be able to calculate the loads transmitted from the roof and the snow to the purlins.

    • In week $5$, we will be able to identify the load carried by each truss element.

    • Operations such as addition of forces are essential for these calculations.

  • Understanding Force Components in Trusses (e.g., Member CD):

    • The force in member CD has two components, one in the x-direction and the other in the y-direction (AA).

    • The two components of the force in CD are orthogonal, meaning they have different directions. Thus, arithmetic addition does not work; their directions must be considered (BB).

    • Therefore, the correct statement is (AA) and (BB).

  • Vector Operations Review in Structural Context: Structural elements like hooks can fail due to combined forces (extF<em>1ext{F}<em>1 and extF</em>2ext{F}</em>2), necessitating vector addition to understand the total effect.

2.1. Scalars and Vectors

Scalars

  • Examples: Mass, Volume

  • Characteristics: It has a magnitude (positive or negative).

  • Addition Rule: Simple arithmetic (e.g., $2 ext{ kg} + 3 ext{ kg} = 5 ext{ kg}).

  • Special Notation: None.

Vectors

  • Examples: Force, Velocity

  • Characteristics: It has both a magnitude and a direction.

  • Addition Rule: Follows the Parallelogram Law.

  • Special Notation: Bold font ($\mathbf{A}$), a line above ($\bar{A}$), or an arrow above ($\vec{A}$).

Vector Operations

1. Parallelogram Law (Tail-to-Tail)

  • To add two vectors, say \mathbf{A}andand\mathbf{B},placetheirtailsatthesamepoint.</p></li><li><p>Completeaparallelogramwithsidesparalleltothevectors.</p></li><li><p>Thediagonaloftheparallelogram,startingfromthecommontail,representstheresultantvector, place their tails at the same point.</p></li><li><p>Complete a parallelogram with sides parallel to the vectors.</p></li><li><p>The diagonal of the parallelogram, starting from the common tail, represents the resultant vector\mathbf{R}.</p></li><li><p>Equation:.</p></li><li><p>Equation:\mathbf{R} = \mathbf{A} + \mathbf{B}</p></li></ul><h5id="5dcc834d−bcbd−4af2−a657−bd744aafebb9"data−toc−id="5dcc834d−bcbd−4af2−a657−bd744aafebb9"collapsed="false"seolevelmigrated="true">2.TriangleMethod(Tip−to−Tail)</h5><ul><li><p>Toaddtwovectors</p></li></ul><h5 id="5dcc834d-bcbd-4af2-a657-bd744aafebb9" data-toc-id="5dcc834d-bcbd-4af2-a657-bd744aafebb9" collapsed="false" seolevelmigrated="true">2. Triangle Method (Tip-to-Tail)</h5><ul><li><p>To add two vectors\mathbf{A}andand\mathbf{B},placethetailof, place the tail of\mathbf{B}atthetipofat the tip of\mathbf{A}.</p></li><li><p>Theresultantvector.</p></li><li><p>The resultant vector\mathbf{R}isdrawnfromthetailofis drawn from the tail of\mathbf{A}tothetipofto the tip of\mathbf{B}.</p></li><li><p>Equation:.</p></li><li><p>Equation:\mathbf{R} = \mathbf{A} + \mathbf{B}(or(or\mathbf{R} = \mathbf{B} + \mathbf{A},demonstratingthecommutativepropertyofvectoraddition).</p></li></ul><h5id="8e5d1da6−d4ca−4204−9359−7839882b23b3"data−toc−id="8e5d1da6−d4ca−4204−9359−7839882b23b3"collapsed="false"seolevelmigrated="true">TrigonometricApproachforVectorAddition</h5><p>Whentheparallelogramortrianglemethodisused,themagnitudeanddirectionoftheresultantvectorcanbefoundusingtheLawofCosinesandtheLawofSines.</p><ul><li><p><strong>CosineLaw:</strong>Tofindthemagnitudeoftheresultant, demonstrating the commutative property of vector addition).</p></li></ul><h5 id="8e5d1da6-d4ca-4204-9359-7839882b23b3" data-toc-id="8e5d1da6-d4ca-4204-9359-7839882b23b3" collapsed="false" seolevelmigrated="true">Trigonometric Approach for Vector Addition</h5><p>When the parallelogram or triangle method is used, the magnitude and direction of the resultant vector can be found using the Law of Cosines and the Law of Sines.</p><ul><li><p><strong>Cosine Law:</strong> To find the magnitude of the resultant\mathbf{C}whentwosides(when two sides (\mathbf{A}, \mathbf{B})andtheangle() and the angle (c)betweenthemareknown:<br>) between them are known:<br>C = \sqrt{A^2 + B^2 - 2AB \cos c}</p></li><li><p><strong>SineLaw:</strong>Tofindangleswhensidesandanangleareknown:<br></p></li><li><p><strong>Sine Law:</strong> To find angles when sides and an angle are known:<br>\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}<br>(Where<br>(Wherea, b, caretheanglesoppositesidesare the angles opposite sidesA, B, Crespectivelyinaforcetriangle).</p></li><li><p><strong>ConnectiontoPythagoreanTheorem:</strong>ThePythagoreanTheoremisaspecialcaseoftheCosineLawwhentheanglerespectively in a force triangle).</p></li><li><p><strong>Connection to Pythagorean Theorem:</strong> The Pythagorean Theorem is a special case of the Cosine Law when the anglecisis90^\circ,as, as\cos 90^\circ = 0,simplifyingtheformulato, simplifying the formula toC = \sqrt{A^2 + B^2}.</p></li></ul><h5id="4bcfca6d−5617−4357−a870−a82885a1e83a"data−toc−id="4bcfca6d−5617−4357−a870−a82885a1e83a"collapsed="false"seolevelmigrated="true">CollinearVectors</h5><ul><li><p>Whenvectorsarecollinear(actalongthesameline),theParallelogramLawsimplifiestoscalar(algebraic)addition.</p></li></ul><h5id="f464150f−935a−4099−b814−cbe2731c578c"data−toc−id="f464150f−935a−4099−b814−cbe2731c578c"collapsed="false"seolevelmigrated="true">VectorSubtraction</h5><ul><li><p>Vectorsubtractionisdefinedastheadditionofanegativevector:<br>.</p></li></ul><h5 id="4bcfca6d-5617-4357-a870-a82885a1e83a" data-toc-id="4bcfca6d-5617-4357-a870-a82885a1e83a" collapsed="false" seolevelmigrated="true">Collinear Vectors</h5><ul><li><p>When vectors are collinear (act along the same line), the Parallelogram Law simplifies to scalar (algebraic) addition.</p></li></ul><h5 id="f464150f-935a-4099-b814-cbe2731c578c" data-toc-id="f464150f-935a-4099-b814-cbe2731c578c" collapsed="false" seolevelmigrated="true">Vector Subtraction</h5><ul><li><p>Vector subtraction is defined as the addition of a negative vector:<br>\mathbf{R'} = \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})</p></li><li><p>Thenegativeofavector(</p></li><li><p>The negative of a vector (-\mathbf{B})hasthesamemagnitudeastheoriginalvectorbutpointsintheoppositedirection.</p></li></ul><h5id="3fceb5cf−eae2−404b−a8ad−dff50d3b6449"data−toc−id="3fceb5cf−eae2−404b−a8ad−dff50d3b6449"collapsed="false"seolevelmigrated="true">ScalarMultiplication</h5><ul><li><p>Multiplyingavectorbyapositivescalarchangesitsmagnitudebutnotitsdirection(e.g.,) has the same magnitude as the original vector but points in the opposite direction.</p></li></ul><h5 id="3fceb5cf-eae2-404b-a8ad-dff50d3b6449" data-toc-id="3fceb5cf-eae2-404b-a8ad-dff50d3b6449" collapsed="false" seolevelmigrated="true">Scalar Multiplication</h5><ul><li><p>Multiplying a vector by a positive scalar changes its magnitude but not its direction (e.g.,2\mathbf{A},whichdoublesthemagnitudeof, which doubles the magnitude of\mathbf{A}).</p></li><li><p>Multiplyingavectorbyanegativescalarchangesitsmagnitudeandreversesitsdirection(e.g.,).</p></li><li><p>Multiplying a vector by a negative scalar changes its magnitude and reverses its direction (e.g.,-0.5\mathbf{A},whichhalvesthemagnitudeof, which halves the magnitude of\mathbf{A}andpointsitintheoppositedirection).</p></li></ul><h4id="f95d829f−0538−4a14−a61e−a405d6719403"data−toc−id="f95d829f−0538−4a14−a61e−a405d6719403"collapsed="false"seolevelmigrated="true">2.3.VectorAdditionofForces</h4><h5id="985bd36b−4138−42c6−95d4−50ffd6868923"data−toc−id="985bd36b−4138−42c6−95d4−50ffd6868923"collapsed="false"seolevelmigrated="true">A)FindingResultantForces</h5><ul><li><p>Theresultantforceand points it in the opposite direction).</p></li></ul><h4 id="f95d829f-0538-4a14-a61e-a405d6719403" data-toc-id="f95d829f-0538-4a14-a61e-a405d6719403" collapsed="false" seolevelmigrated="true">2.3. Vector Addition of Forces</h4><h5 id="985bd36b-4138-42c6-95d4-50ffd6868923" data-toc-id="985bd36b-4138-42c6-95d4-50ffd6868923" collapsed="false" seolevelmigrated="true">A) Finding Resultant Forces</h5><ul><li><p>The resultant force\mathbf{FR}oftwoforcesof two forces\mathbf{F}1andand\mathbf{F}2istheirvectorsum:is their vector sum:\mathbf{FR} = \mathbf{F}1 + \mathbf{F}2</p></li><li><p>ThiscanberepresentedgraphicallyusingtheParallelogramLaworTriangleMethod.</p></li></ul><h5id="53146ac4−1cf1−4ca6−8b0c−99909efe2c36"data−toc−id="53146ac4−1cf1−4ca6−8b0c−99909efe2c36"collapsed="false"seolevelmigrated="true">B)FindingtheComponentsofaForce:VectorResolution</h5><ul><li><p><strong>Definition:</strong>Vectorresolutionistheprocessofbreakingupavectorintoitscomponentsalongspecifiedaxes(e.g.,xandyaxes).</p></li><li><p>Thisisessentiallythereverseprocessoffindingaresultantforcefromitscomponents,alsobasedontheParallelogramLaw.</p></li></ul><h5id="bcbe0aff−5294−4d59−b842−09e9e5bc0c02"data−toc−id="bcbe0aff−5294−4d59−b842−09e9e5bc0c02"collapsed="false"seolevelmigrated="true">C)AdditionofSeveralForces</h5><ul><li><p>Toaddmorethantwoforces(e.g.,</p></li><li><p>This can be represented graphically using the Parallelogram Law or Triangle Method.</p></li></ul><h5 id="53146ac4-1cf1-4ca6-8b0c-99909efe2c36" data-toc-id="53146ac4-1cf1-4ca6-8b0c-99909efe2c36" collapsed="false" seolevelmigrated="true">B) Finding the Components of a Force: Vector Resolution</h5><ul><li><p><strong>Definition:</strong> Vector resolution is the process of breaking up a vector into its components along specified axes (e.g., x and y axes).</p></li><li><p>This is essentially the reverse process of finding a resultant force from its components, also based on the Parallelogram Law.</p></li></ul><h5 id="bcbe0aff-5294-4d59-b842-09e9e5bc0c02" data-toc-id="bcbe0aff-5294-4d59-b842-09e9e5bc0c02" collapsed="false" seolevelmigrated="true">C) Addition of Several Forces</h5><ul><li><p>To add more than two forces (e.g.,\mathbf{F}1, \mathbf{F}2, \mathbf{F}3),theycanbeaddedsequentially(tip−to−tailmethodrepeatedly):), they can be added sequentially (tip-to-tail method repeatedly):\mathbf{FR} = (\mathbf{F}1 + \mathbf{F}2) + \mathbf{F}_3</p></li><li><p>Thefinalresultantforce</p></li><li><p>The final resultant force\mathbf{F_R}producesthesameeffectasallindividualforcesactingtogether.</p></li></ul><h4id="f233ea7d−1780−4235−a470−556e6079bdb0"data−toc−id="f233ea7d−1780−4235−a470−556e6079bdb0"collapsed="false"seolevelmigrated="true">2.4.AdditionofaSystemofCoplanarForces</h4><h5id="809008f8−55c8−4408−988f−f4683f8d53a3"data−toc−id="809008f8−55c8−4408−988f−f4683f8d53a3"collapsed="false"seolevelmigrated="true">A)ScalarNotation</h5><ul><li><p><strong>RectangularComponents:</strong>Foraforceproduces the same effect as all individual forces acting together.</p></li></ul><h4 id="f233ea7d-1780-4235-a470-556e6079bdb0" data-toc-id="f233ea7d-1780-4235-a470-556e6079bdb0" collapsed="false" seolevelmigrated="true">2.4. Addition of a System of Coplanar Forces</h4><h5 id="809008f8-55c8-4408-988f-f4683f8d53a3" data-toc-id="809008f8-55c8-4408-988f-f4683f8d53a3" collapsed="false" seolevelmigrated="true">A) Scalar Notation</h5><ul><li><p><strong>Rectangular Components:</strong> For a force\mathbf{F}inarighttriangle,itsrectangularcomponentsin a right triangle, its rectangular components\mathbf{Fx}andand\mathbf{Fy}aredeterminedusingtrigonometry:</p><ul><li><p>are determined using trigonometry:</p><ul><li><p>F_x = F \cos \theta</p></li><li><p></p></li><li><p>F_y = F \sin \theta<br>(Where<br>(Where\thetaistheangletheforceis the angle the force\mathbf{F}$$ makes with the positive x-axis).

  • **Using a