Understanding Forces: Types and Characteristics

Concepts and Principles of Force

• Force describes the action exerted by one body on another.
• Force tends to change the state of rest or motion of a body.

Characteristics of a Force

• Magnitude: The quantitative effect of a force.
• Point of Application: The exact location where a force is applied.
• Direction: Where the force moves along the line of action.

Types of Force Systems

• Concurrent Forces: Forces whose lines of action pass through one common point.
• Non-Concurrent Forces: Forces where the lines of action do not meet at a common point.
• Collinear Forces: Forces where the lines of action act along the same line.
• Parallel Forces: Forces in the same direction whose lines of action never meet.
• Coplanar Force: Where the lines of action lie in a single plane.
• Non-Coplanar Force: Where the lines of action do not lie in one plane.

Scalar and Vector Quantities

• Scalar Quantity: A quantity completely specified by a single value with an appropriate unit and no direction.
• Vector Quantity: A quantity with both numerical and directional properties.

Vectors

• Vectors represent physical quantities with magnitude and direction.
• Vectors in text or symbolic names are typeset in bold (e.g., F).
• Common examples of vectors displacement, velocity, weight, moment and acceleration.

Vector Components

• x-component of a vector (Aₓ) is the projection along the x-axis, defined as |A|cosθ.
• y-component of a vector (Aᵧ) is the projection along the y-axis, defined as |A|sinθ.
• The magnitude of a vector is given by √(|Aₓ|² + |Aᵧ|²)
• The angle θ of the vector with the x-axis is found using tanθ = |Aᵧ|/|Aₓ|.

Vector Magnitude and Unit Vectors

• Vector Magnitude: Positive real number including units which describes the intensity or strength of the vector.
• Unit Vectors: Vectors with a magnitude of 1 and no units, used to describe a direction in space.
  • x-component is i
  • y-component is j.
• Vectors A and B can be expressed in terms of unit vectors, like:
  • A = Aₓi + Aᵧj
  • B = Bₓi + Bᵧj

Two-Dimensional Vector Form

• Can be written as A = (a₁, a₂).
• Vector in component form: xi + yj or (x, y).

Three-Dimensional Vector Form

• Can be written as A = (a₁, a₂, a₃).
• Vectors A in component form: xi + yj + zk or (x, y, z).

Vector Addition

• Vectors can be added using the Parallelogram Law or Triangle Rule.
• Polygon Law of Addition allows to iteratively add additional vectors
• For vectors A and B, the resultant R = A + B

Vector Subtraction

• Corresponds to the addition of a negative vector.
• To subtract B from A, add the negative of vector B to vector A : A - B = A + (-B) = R

Resultant Force

• The resultant force describes the net force acting on an object.

• Calculating the Sum of Forces:

  • Forces at an angle from the coordinate axes can be resolves into mutually perpendicular components
  • The sum of the force in the x direction, Fₓ is given by Fₓ = cos θ
  • The sum of the force in the y direction, Fᵧ is given by Fᵧ = sin θ
  • The value of F is given by F = √(ΣFₓ)² + (ΣFᵧ)²
  • The value of θ is given by tanθ = ΣFᵧ/ΣFₓ

Dot Product

• Dot product is the sum of the products of corresponding components.
• Referred to as scalar product. Vector dot product is expressed as
  • A · B = (AₓBₓ) + (AᵧBᵧ) + (A₂B₂)
  • A · B = |A||B|cosθ

Cross Product

• The cross product of two vectors A and B results in a third vector C that is perpendicular to both A and B.
• Formula for calculating cross product
  • A x B = [i(b₁ c₂ - b₂c₁) - j(a₁c₂ – a₂c₁) + k (a₁b₂- a₂b₁)]
  • A x B = |A||B|sinθ