3.2.5 F = ma - Vector Notation

How is Newton’s Second Law (N2L) used with vectors?

• The resultant force (F) and acceleration (a) are vectors
  • For forces and motion in two dimensions, F N and a m s^-2 will be made up of two components – a horizontal (x-) component and a vertical (y -) component
• Displacement , velocity and weight are also vector quantities
• Time and mass are scalar quantities
• Vectors appear in bold (non-italic) font in textbooks, on exam papers, etc (i.e. F, a) but in handwriting should be underlined (i.e. F , a )

What notation is used for forces as vectors?

• All vectors are written either as column vectors or in i-j format
• As a column vector would look like
  

                                                         

• In i-j notation   would look like
                                             

When do I use F = ma (N2L) in vector/2D form?

• If vectors/2D are being used this will be clear from the information given in the question – any vector quantities will be given as a column vector or written in i-j notation
• Remember  is used when motion is involved – equations may come from ‘suvat’ (if the acceleration is constant), or using N2L directly; look for (resultant) force, mass and acceleration being involved
• Use (N2L in 1D) or an appropriate ‘’ (in 1D) equation to set up and solve separate equations for both the horizontal (-) and vertical (-) components.

How is Newton’s Second Law (N2L) used with problems involving weight?

• Weight is a force, so it is a vector quantity
  • where  is the acceleration due to gravity
• Weight always acts vertically downwards so it only acts in the j-direction

                                                                

• Treating the two dimensions separately means weight only needs to be considered when looking at the vertical ( -) direction
• Most 2D/vector problems are based on a bird’s-eye view – the two dimensions being left/right and forwards/backwards, so the up/down (third) dimension where weight would apply, is often not involved

How do we apply Newton’s Second Law (F = ma) in problems involving vectors?

• Step 1. If necessary, draw a diagram and label all forces acting on the particle(s)
  • label the i and j directions and any other useful information.
  • If a diagram is given, add any missing information to it.
• Step 2. Taking each dimension/component at a time use F = ma
  • If there is more than one particle involved you may have to do this for each
• Step 3. Solve the equations for each component and put the final answers back into vector notation
  • In some harder problems simultaneous equations may arise

 

Worked Example

Exam Tip