Non-unform Objects & Other Problems (Edexcel International A Level Maths: Mechanics 2)

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Non-unform Objects & Other Problems

What is meant by a non-uniform lamina/framework?

  • Strictly speaking a non-uniform lamina would be one that has different densities at different points on its surface and a non-uniform framework would be one that is made from rods/wires that have different densities along their lengths.
    • e.g. a plank of wood (timber) may have knots in places on its surface
  • However, for the problems you encounter in IAL Mechanics 2 non-uniform means each individual lamina making a composite lamina may be of a different density but each individual lamina is uniform
  • Similarly, for a framework, each individual rod/wire will be uniform, but the framework may be made from rods/wires of different densities.

What other problems may I encounter in IAL Mechanics 2 (M2)?

  • Non-uniform problems
    • Non-uniform (composite) laminas may be created by having uniform laminas overlapping each other or by taking part of a lamina and folding it back upon itself
    • Non-uniform frameworks would be made from rods/wires that have different (but related) densities or thickness
    • The relationship between the different densities will be explained in a question
      • e.g. part of a (uniform) lamina folded back on itself creates an area of double density
      • e.g. one cable in a framework is twice the thickness of the others
    • In solving problems remember, due to uniformity, we use can use area (laminas) and length (frameworks) rather than mass – so if we have double density, we double the area/length where necessary
  • Another type of problem is where a lamina or a framework have an object (or masses) attached to them.
    • This will affect the position of the centre of mass of the lamina or framework
      • e.g. a framework made from metal rods could be holding up spotlights above a stage and so have lights (masses) attached to some (or all) rods
  • Problems may involve laminas or frameworks being suspended by more than one string to keep them in a particular position
    • e.g. A sign hanging from a ceiling that needs to be pointing directly left
    • Such problems normally involved finding the position of the centre of mass and/or the tension in the strings, using skills from various parts of the mechanics course

2-1-7-fig6-other-problems

Worked example

A rectangular uniform lamina has one edge folded back upon itself as shown in the diagrams below.

2-1-7---we-diagram

(a)
Find the coordinates of the centre of mass of the folded lamina

 

(b)
The folded lamina is suspended from a ceiling by two strings attached at the points P and Q. The folded lamina remains horizontal in equilibrium. Find the tension in the two strings.
(a)
Find the coordinates of the centre of mass of the folded lamina

2-1-7-fig7-we-solution_a

(b)
The folded lamina is suspended from a ceiling by two strings attached at the points P and Q. The folded lamina remains horizontal in equilibrium. Find the tension in the two strings.
2-1-7-fig7-we-solution_b

Examiner Tip

  • Ensure you know whether you are working with a lamina or a framework.
  • Folded and overlapping laminas may form shapes that have axes of symmetry which can cut down the work in finding the position of the centre of mass.

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.