Resolving Vectors (AQA AS Physics)

Revision Note

Lindsay Gilmour

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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

Representing Vectors, downloadable AS & A Level Physics revision notes

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

Resolving Vectors, downloadable AS & A Level Physics revision notes

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

  • For the horizontal component, Fx = F cos θ
  • For the vertical component, Fy = 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)

Vectors On an Inclined Plane, downloadable AS & A Level Physics revision notes

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

Worked example

A helicopter provides a lift of 250 kN when the blades are tilted at 15º from the vertical.Resolving Forces Worked Example, downloadable AS & A Level Physics revision notesCalculate the horizontal and vertical components of the lift force.

Step 1: Draw a vector triangle of the resolved forces

4.1.2 Resolving Forces Worked Example Answer

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

Examiner Tip

If you're unsure as to which component of the force is cos θ or sin θ, just remember that the cos θ is always the adjacent side of the right-angled triangle AKA, making a 'cos sandwich'Resolving Vectors Exam Tip, downloadable AS & A Level Physics revision notes

Equilibrium

  • 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:

Vector triangle in equilibrium, downloadable AS & A Level Physics revision notes

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

Worked example

A weight hangs in equilibrium from a cable at point X. The tensions in the cables are T1 and T2 as shown.WE - Forces in equilibrium question image 1, downloadable AS & A Level Physics revision notesWhich diagram correctly represents the forces acting at point X?WE - Forces in equilibrium question image 2, downloadable AS & A Level Physics revision notes

Equilibrium Worked Example (3), downloadable AS & A Level Physics revision notes

Examiner Tip

The diagrams in exam questions about this topic could ask you to draw to scale, so make sure you have a ruler handy!

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Lindsay Gilmour

Author: Lindsay Gilmour

Expertise: Physics

Lindsay graduated with First Class Honours from the University of Greenwich and earned her Science Communication MSc at Imperial College London. Now with many years’ experience as a Head of Physics and Examiner for A Level and IGCSE Physics (and Biology!), her love of communicating, educating and Physics has brought her to Save My Exams where she hopes to help as many students as possible on their next steps.