Particles (Edexcel International A Level Maths: Mechanics 2)

What is/are centres of mass?

• The centre of mass of a body (or a system of bodies) is the point at which the total mass of the body (or system) can be considered to act as one
  • It is the single point at which the (force) weight ( W = mg)of the whole system acts
• Bodies are modelled as particles

What is meant by particles along a straight line?

• This is when several particles making a system are arranged in a straight line
• This will be either horizontally or vertically – usually with reference to the x-axis or y-axis
  •  would be the coordinates of the centre of mass along the x-axis
  • would be the coordinates of the centre of mass along the y-axis

So the particles do not have to be connected to each other?

• Not necessarily.
• Any object that is used to connect the particles can affect the centre of mass (see Revision Note 2.1.7 Non-uniform Objects & Other Problems) unless it is modelled as being light
  • e.g. Tins of paint (particles) placed at various points along a plank of wood (connecting object)
    • If the plank of wood is modelled as light it will have no influence on the position of the centre of mass – i.e. it can be ignored

How do I find the centre of mass of a system of particles along a straight line

• Firstly, if no axis or origin have been given in a question, you will need to create your own
  • For example, the left-hand edge of the plank of wood could be the origin with the -axis increasing towards the right-hand edge of the plank of wood

• In general for a system of n particles with masses placed along the x-axis at the points with coordinates , the centre of mass, point ( , 0) is found by solving

• We’re sure you can figure out the equivalent equation for particles placed along the y-axis!

STEP 1        Draw a diagram indicating the particles and their positions on the line.

                     This does not need to be a scale diagram but should indicate the axis and origin.

                     (Remember you may have to create your own axis and origin.)

                     If given a diagram, add anything necessary to it.

                       

STEP 2       Set up an equation for the -coordinate for the centre of mass using

                    (This is the same equation as above but you may sometimes see

                    it with the n  and the i =1 removed, as we have done here.)

STEP 3       Solve the equation and answer the question.

         This may be giving the centre of mass as coordinates or describing its position relative to an object or body.

                   You can check your answer is sensible by comparing it to the location of the particles.

What is meant by particles in a (2D) plane?

• Particles in a (2D) plane refers to a system where particles are arranged on a surface
  • e.g. The playing pieces (particles) on a chess board (plane)
• The two dimensions are horizontal and vertical
  • Cartesian (x -y axes) coordinates are used to describe the positions of the particles with  being the coordinates of the centre of mass of the system
• If the particles are connected by an object – including the plane itself - the centre of mass may be affected (see Revision Note 2.1.7 Non-uniform Objects & Other Problems) - unless the connecting object is modelled as being light

How do I find the centre of mass of a system of particles in a (2D) plane?

• Firstly, if no axis or origin have been given in a question, you will need to create your own
  • For example, the bottom left corner of a sheet of metal edge could be the origin with the x-axis running along the bottom edge of the sheet and the y-axis running up the left hand side of the sheet.

• In general for a system of n particles with masses  placed in a 2D plane at the points with coordinates , the centre of mass, point  is found by solving

where are the position vectors of the particles and
  is the position vector of the centre of mass

• (Position) vectors can be given as column vectors; or in i-j  notation; 
• Alternatively the two dimensions can be separated creating two systems of particles in a straight line
  • Use the equations   and    to find the coordinates of the centre of mass, separately
  • Remember to give your final answer in the form 

STEP 1       Draw a diagram indicating the particles and their positions on the plane.

                    This does not need to be a scale diagram but should indicate the axes and origin.

                    (Remember you may have to create your own axes and origin.)

                    If given a diagram, add anything necessary to it.

STEP 2       Set up an equation for the position vector of the centre of mass using

(This is the same equation as above but you may sometimes see it with the n  and th   i =1 removed, as we have done here.)

                     Alternatively, you can set up separate equations for  and .

STEP 3        Solve the equation(s) and answer the question.

This may be giving the position of the centre of mass as coordinates, a position vector or describing its position relative to an object or body (including the plane).

You can check your answer is sensible by comparing it to the location of the particles.