4.1.2 Resolving Vectors

• Two vectors can be represented by a single resultant vector
  • Resolving a vector is the opposite of adding vectors

• A single resultant vector can be resolved
  • This means it can be represented by two vectors, which in combination have the same effect as the original one

• When a single resultant vector is broken down into its parts, those parts are called components
• For example, a force vector of magnitude F and an angle of θ to the horizontal is shown below

The resultant force F at an angle θ to the horizontal

• It is possible to resolve this vector into its horizontal and vertical components using trigonometry

The resultant force F can be split into its horizontal and vertical components

• For the horizontal component, F_x = F cos θ
• For the vertical component, F_y = F sin θ

Forces on an Inclined Plane

• Objects on an inclined plane is a common scenario in which vectors need to be resolved
  • An inclined plane, or a slope, is a flat surface tilted at an angle, θ

• Instead of thinking of the component of the forces as horizontal and vertical, it is easier to think of them as parallel or perpendicular to the slope
• The weight of the object is vertically downwards and the normal (or reaction) force, R is always vertically up from the object
• The weight W is a vector and can be split into the following components:
  • W cos (θ) perpendicular to the slope
  • W sin (θ) parallel to the slope

• If there is no friction, the force W sin (θ) causes the object to move down the slope
• The object is not moving perpendicular to the slope, therefore, the normal force R = W cos (θ)

The weight vector of an object on an inclined plane can be split into its components parallel and perpendicular to the slope

Step 1: Draw a vector triangle of the resolved forces

Step 2: Calculate the vertical component of the lift force

Vertical = 250 × cos(15) = 242 kN

Step 3: Calculate the horizontal component of the lift force

Horizontal = 250 × sin(15) = 64.7 kN

• Coplanar forces can be represented by vector triangles
• Forces are in equilibrium if an object is either
  • At rest
  • Moving at constant velocity

• In equilibrium, coplanar forces are represented by closed vector triangles
  • The vectors, when joined together, form a closed path

• The most common forces on objects are
  • Weight
  • Normal reaction force
  • Tension (from cords and strings)
  • Friction

• The forces on a body in equilibrium are demonstrated below:

Three forces on an object in equilibrium form a closed vector triangle