Frameworks (Edexcel International A Level Maths): Revision Note

Frameworks

A brief reminder of centre of mass …

• The centre of mass is the point at which the total mass of a body (or a system of bodies) can be considered to act as one

What is a framework?

• A framework is created by joining thin-linear-shaped bodies (e.g. a pole or post) together to create a flat design or shape

• Such bodies can be modelled mathematically as rods or wires

• Examples include

  • the supporting structure for the roof of a house

  • four straight rope lights joined together to make a rectangular rope light

• In Mechanics 2 both uniform (this Revision Note) and non-uniform rods (Revision Note 2.1.7) are considered

• A framework would not be split into standard shapes as perimeter rather than the area is being worked with

What modelling assumptions are used with frameworks?

• (For this Revision Note) all rods (or wires) making a framework are made from the same uniform material

  • uniform means the mass per unit length ( kg) is equal at every point along a rod

    • the mass of a rod is proportional to its length ( m)

    • i.e. Mass of a rod = kg

  • every rod/wire being made from the same material means that the mass per unit length will be the same for every rod/wire

    • i.e. will be the same for every rod

    • ’s cancel in the equation for finding the position of the centre of mass

    • only the values of (i.e. the lengths of each rod) are needed in calculations

• A framework is flat so is modelled as existing in a (2D) plane

  • the third dimension will be small compared to the other two so is negligible

• Any material/mass used in the process to join rods (or wires) is negligible

How do I find the centre of mass of a framework?

• In short, find the length and position of the centre of mass for each individual rod, then treating these each as a particle with mass equal to the length of the rod, use the equation  to find the position vector of the centre of mass of the framework (Revision Note 2.1.1) 

STEP 1   Sketch a diagram, or add to a diagram if one has been given

      Create your own axes if necessary

      Identify the different rods making up the framework

      e.g.

STEP 2   Find the length and the position of the centre of mass of each rod using midpoints

                          List the results in a table to make the equation easier in the next step

     The length of the diagonal rod (Rod 3) is found by Pythagoras’ Theorem,

STEP 3   Treat the midpoints as the positions of particles with masses the length of the rods.

      Use to find the position vector of the centre of mass (G) of the framework.

      Remember to give the final answer in the required format.

      e.g. 

      So the coordinates of the centre of mass (G) of the framework are (4, 7)

                          Looking back at the diagram this seems a sensible answer.

• In the example above  and  could’ve been treated separately rather than use vector notation

What if a rod or wire is curved (circular)?

• In Mechanics 2 if a curved rod or wire is involved in a framework it will be in the shape of a circular arc

• Recall that the formulae for the length of an arc is “” where l is the length of the arc, r is the radius and θ is the angle at the centre measured in radians

• The formula for finding the position of the centre of mass of a circular arc is similar but different to that for a sector of a circle lamina

• This is given in the formulae booklet

• Some questions may work with a circular arc only, some may work a sector of a circle that would include the two radii