Engineering Mechanics: Statics ENG2008 - Topic 2

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Questions and Answers

What is a key method for adding two vectors graphically?

  • Using the Pythagorean theorem only
  • Using the triangle method or the parallelogram law (correct)
  • Using scalar multiplication only
  • Using the cosine rule exclusively

In Cartesian vector notation, how are vector components represented?

  • Only magnitudes without directions
  • Using unit vectors with magnitude and direction (correct)
  • Using vectors without resolving into components
  • Using angles only without magnitudes

Which formula is used to resolve a vector into its components?

  • $a^2 = b^2 + c^2 + 2bc \, cosA$
  • $a^2 = b^2 + c^2 - 2bc \, cosA$ (correct)
  • $a = b + c$
  • None of the above

What does the resolution of a vector involve?

<p>Breaking a vector into its x and y components (B)</p> Signup and view all the answers

Which of the following is NOT a method to add two vectors?

<p>Geometric averaging (A)</p> Signup and view all the answers

What represents the direction in Cartesian vector notation?

<p>Terms i and j (C)</p> Signup and view all the answers

What defines the process of adding multiple concurrent vectors?

<p>Graphical methods or component addition can be utilized (B)</p> Signup and view all the answers

What aspect of the Cartesian system is crucial in vector resolution?

<p>The axes must be perpendicular to each other (B)</p> Signup and view all the answers

What is the first step in the process of adding several vectors?

<p>Resolve each force into its components. (C)</p> Signup and view all the answers

After resolving the forces into components, what should be done next?

<p>Add the x components together and add the y components together. (A)</p> Signup and view all the answers

How is the magnitude of the resultant vector calculated from its components?

<p>Using the formula $F = \sqrt{(FR_x^2 + FR_y^2)}$. (C)</p> Signup and view all the answers

What represents collinear vectors?

<p>Vectors that are parallel to each other. (A)</p> Signup and view all the answers

What is the definition of concurrent vectors?

<p>Vectors whose lines of action intersect. (D)</p> Signup and view all the answers

If the resultant components are FR = { 16.82 i + 3.49 j } kN, what is the angle of the resultant vector approximately?

<p>11.7° (B)</p> Signup and view all the answers

Which of the following is NOT a step in the process of finding the resultant vector?

<p>Add the resultant angles together. (D)</p> Signup and view all the answers

If you are given three forces, how should you categorize them before solving for the resultant vector?

<p>Determine if they are concurrent, collinear, or coplanar. (A)</p> Signup and view all the answers

How is the position vector rAB defined between two points A and B in 3-D space?

<p>rAB = {(XB - XA) i + (YB - YA) j + (ZB - ZA) k} m (D)</p> Signup and view all the answers

What is the correct formula for calculating the magnitude of a vector A = (AX i + AY j + AZ k) m?

<p>A = ((A')^2 + AZ^2)^(1/2) (A), A = (AX^2 + AY^2 + AZ^2)^(1/2) (C)</p> Signup and view all the answers

If two forces F1 and F2 are represented as F1 = {10 i + 20 j} N and F2 = {20 i + 20 j} N, what is the resultant force vector?

<p>{30 i + 40 j} N (D)</p> Signup and view all the answers

What is the angle $\theta$ from the positive x-axis if the resultant vector has components -162.8 i and -521 j?

<p>253° (B)</p> Signup and view all the answers

When resolving a force F along the x and y axes, which of the following is the correct expression for F = 80 N at an angle of 30°?

<p>80 cos(30°) i - 80 sin(30°) j (C)</p> Signup and view all the answers

In Cartesian vector addition, what is the first step to combine vector A = (AX i + AY j + AZ k) m and vector B = (BX i + BY j + BZ k) m?

<p>Combine the i, j, and k components separately. (A)</p> Signup and view all the answers

When deriving the projection of a vector A in the x-y plane, which operation is performed on components AX and AY?

<p>A' = (AX^2 + AY^2)^(1/2) (A)</p> Signup and view all the answers

What is the resultant magnitude of the vector FR = { -162.8 i - 521 j } N?

<p>546 N (D)</p> Signup and view all the answers

Flashcards

What is a vector?

A vector is a quantity that has both magnitude (size) and direction. Examples: Velocity, force, displacement, and acceleration.

What is a scalar?

A scalar is a quantity that only has magnitude (size). Examples: Speed, mass, time, and temperature.

Parallelogram Law

The parallelogram law is a graphical method for adding two vectors by creating a parallelogram with the vectors as adjacent sides. The diagonal of the parallelogram represents the resultant vector.

Triangle Method

The triangle method is a graphical method for adding two vectors by placing the tail of one vector at the head of the other. The resultant vector is drawn from the tail of the first vector to the head of the second vector.

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Resolution of a Vector

The process of breaking down a vector into its component parts along the x-axis and y-axis.

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Cartesian Vector Notation

A standard way to represent vectors using unit vectors 'i' and 'j' along the x and y axes, respectively. Each component of the vector is shown as a magnitude and a direction ('i' or 'j'). Example: F = Fx i + Fy j”,

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Resultant Vector

The sum of two or more vectors. It can be found graphically using the parallelogram or triangle method, or analytically using Cartesian Vector Notation.

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Resolving a Vector

A process of breaking down a vector into its horizontal (x) and vertical (y) components.

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Collinear Vectors

Vectors that have the same line of action.

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Concurrent Vectors

Vectors that intersect at a single point.

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Coplanar Vectors

Vectors that lie in the same plane.

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Finding Magnitude and Angle of Resultant Vector

The process of finding the magnitude (length) and direction (angle) of the resultant vector.

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Adding Components of Vectors

Adding the corresponding x-components and y-components of multiple vectors to obtain the x-component and y-component of the resultant vector.

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Position Vector in 3-D

A position vector in 3-D space is a fixed vector that describes the location of a point relative to another point. It's represented by the difference in coordinates between the two points.

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3-D Coordinates (x, y, z)

The coordinates of a point in 3-D space are represented as (x, y, z). These coordinates dictate the position of the point along the x, y, and z axes.

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3-D Cartesian Vector Notation

In 3-D space, vectors are represented using 'i', 'j', and 'k' unit vectors along the x, y, and z axes respectively. Each vector is then described by its magnitude along each axis.

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Magnitude of 3-D Vector

The magnitude of a 3-D position vector is calculated by adding up the squares of each component, then taking the square root. This represents the overall distance of the point from the origin.

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Adding/Subtracting 3-D Vectors

The process of adding or subtracting 3-D vectors is done by adding or subtracting the corresponding components (i, j, and k terms) of each vector.

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Resultant Vector (3-D)

The resultant vector is the sum of multiple vectors. It represents the overall effect of combining these vectors.

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Scalars vs. Vectors (3-D)

A scalar is a quantity with only magnitude (size), while a vector has both magnitude and direction. Vectors in a 3-D space also have a direction in the z-axis (represented by 'k').

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Projecting a 3-D Vector onto x-y Plane

The projection of a 3-D vector onto the x-y plane is a 2-D vector. Its magnitude is calculated using the Pythagorean theorem, considering only the x and y components.

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Study Notes

Engineering Mechanics: Statics ENG2008 - Topic 2

  • Students will be able to resolve a 2-D or 3-D vector into its components.
  • Students will be able to add 2-D or 3-D vectors using Cartesian vector notation.
  • In-class activities include applying vector addition to solving force problems, resolving vectors using Cartesian Vector Notation (CVN), adding vectors using CVN, and an attention quiz.
  • Example forces include F₂ = 150 N with a 10° angle, and F₁ = 100 N with a 15° angle.

Scalars and Vectors

  • Scalars have only magnitude (e.g., mass, volume).
  • Vectors have both magnitude and direction (e.g., force, velocity).
  • Scalars are added using simple arithmetic.
  • Vectors are added using the parallelogram law.
  • Vectors are often represented using bold font, a line, an arrow, or a 'carrot' symbol.

Vector Operations

  • Scalar multiplication and division change the magnitude of a vector, but not its direction.
  • Examples include 2A, -1.5A, 0.5A.

Vector Addition

  • Parallelogram law: Used to add two vectors. The resultant vector is the diagonal of the parallelogram formed by the two vectors.
  • Triangle Method (tip-to-tail): Always used to add vectors. One vector is drawn, and the tail of the next vector starts from the tip of the first vector, and so on. The resultant vector starts from the tail of the first vector to the tip of the last vector.
  • These methods are valid for two vectors, but the triangle method does not need the vectors to be drawn to scale.

Vector Addition (Example)

  • Example given involves concurrent forces acting on a bracket, combining forces with different magnitudes and angles.

Vector Addition (More Than Two Concurrent Vectors)

  • Concurrent vectors, more than two, are combined by resolving all vectors into their components in the X and Y direction/vectors. Adding the i and j components separately will result in the resultant force.

Resolution of a Vector

  • "Resolution" is breaking a vector into components along chosen directions. It's the reverse of the parallelogram law.
  • Resolution allows traveling in determined directions instead of one stage.

Cartesian Vector Notation

  • Vectors are resolved into x and y components (and potentially z) using the axes system.
  • Components are expressed with magnitudes (e.g., Fₓ, Fᵧ) and direction vectors.
  • Note that i and j are the "direction only" unit vectors used for x and y respectively.

Example of Cartesian Vector Notation

  • Cartesian vectors use unit vectors to denote the direction of the components as either positive (i,j,k) or negative (-i, -j, -k)

Application of Vector Addition

  • Example given showing concurrent forces acting on a bracket.

Addition of Several Vectors

  • Step 1: Resolve each force into components.
  • Step 2: Sum the components along the i and j (and k) directions individually.
  • Step 3: Combine resultant components into resultant vector magnitude and angle.
  • Example of calculation process given.

Magnitude and Angle Examples

  • Show examples of calculating resultant magnitude from resolved components in both x and y direction.
  • Show example of calculating angle from the x and y components of the resultant vector.

Example Problem

  • Given three concurrent forces acting on a bracket.
  • Find the magnitude and angle of the resultant force.
  • Plan given for solution:
    • Resolve forces into x-y components
    • Add respective components to get the resultant vector
    • Find magnitude and angle from resultant components

3-D Cartesian Vector Terminology - Position Vectors

  • Position vector: Locates a point in space relative to another point.
  • 3D Cartesian co-ordinates are used for points.

3-D Cartesian Vector Terminology

  • A 3D Cartesian vector can be defined using Cartesian vector notation (x,y,z).
  • Projections onto X-Y plane are combined with the Z-axis component to give the magnitude of the vector.
  • Illustrative example given showing the dimensions needed and how to derive the vector

Addition/Subtraction of Vectors in 3-D

  • Adding or subtracting 3-D vectors involves combining corresponding components (x, y, z).

Important Considerations for 3-D Vectors

  • Sometimes information is given as magnitude and coordinate direction angles or magnitude and projection angles.
  • Vectors can be changed from this information into Cartesian form in different representations.

Unit vector

  • A unit vector has a magnitude of 1, is dimensionless, and points in the same direction as the original vector.
  • The unit vectors in the Cartesian system are i, j, and k, along the positive x, y, and z axes respectively.

End of Lecture

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