AQA A Level Physics

Revision Notes

4.1.2 Resolving Vectors

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 notes

Calculate 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

Exam 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.

Equilibrium Worked Example (1), downloadable AS & A Level Physics revision notes

Which diagram correctly represents the forces acting at point X?

Equilibrium Worked Example (2), downloadable AS & A Level Physics revision notes

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

Exam 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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