Uniform Laminas (Edexcel International A Level Maths: Mechanics 2)

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 lamina?

• A lamina is a (3D) body that has one very small dimension compared to the other two dimensions which means the third dimension is negligible and the body can be modelled as a (2D) plane
  • Common examples include
    • a sheet of paper, card, wood, metal, plastic, cloth, etc
    • a wall

What modelling assumptions are used with laminas?

• (For this Revision Note) all laminas are uniform
  • uniform means the mass per unit area (m kg) is equal at every point on the lamina
    • In the real world a uniform lamina is very difficult to achieve – particularly with natural materials such as timber where knots and other imperfections occur
    • The cloth on a professional snooker table needs to be as close as possible to a uniform lamina - it should have no imperfections (such as a small tear) if the table and balls are to play as a professional would expect
• In Mechanics 2 both uniform (this Revision Note) and non-uniform laminas (Revision Note 2.1.7) are considered

How do I find the centre of mass of a (standard) uniform lamina?

• If the shape of a lamina has an axis of symmetry, the position of the centre of mass will be somewhere on that axis
• If the shape of a lamina has more than one axes of symmetry, the position of the centre of mass will be the intersection of those axes
• The most common shapes – rectangles (including squares), triangles, circles and sectors (including semi-circles) are referred to as standard uniform laminas

How do I find the centre of mass of a triangular lamina?

• Triangles are a little more complicated
• In any triangle a median is a line joining a vertex to the midpoint of the opposite side
• In any uniform triangle the position of the centre of mass
  • lies of the way along a median when measured from its vertex.
    This is stated in the formulae booklet
  • is the point of intersection of the medians

• The coordinates of the centre of mass of any uniform triangle can be found if the coordinates of its three vertices are known
  • The coordinates of the centre of mass of any uniform triangular lamina is the mean of the coordinates of its vertex
  • This only works for triangles and is not true for all other polygons

• In addition to the above, for equilateral and isosceles triangles only
  • In an equilateral triangle all three medians are also axes of symmetry
  • In an isosceles triangle one of the medians is an axis of symmetry
  • The position of the centre of mass will always lie on an axis of symmetry so for both equilateral and isosceles triangles only one median need be used
  • An axis of symmetry is perpendicular to the side they bisect so any missing lengths can be found using Pythagoras’ theorem and/or basic trigonometry