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 F =?ma?would look like

• In?i-j?notation?F = ma?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?F = ma?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?F= ma(N2L in 1D) or an appropriate ‘suvat’ (in 1D) equation to set up and solve?separate?equations for both the?horizontal?(?x-) and?vertical?(y?-) components.

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

• Weight is a?force, so it is a vector quantity
  • W?= mg?N where?g?m s-2?is the?acceleration?due to?gravity
• Weight always acts?vertically?downwards?so it only acts in the?j-direction

? ? ? ? ? ? ? ? ? (g≈ 9.8 m s-2?)

• Treating the two dimensions separately means weight only needs to be considered when looking at the vertical (y?-) 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

A Force, F, acting on a snooker ball

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