Composite Laminas

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 composite lamina?

A composite lamina is a non-standard shaped lamina that can be constructed from two or more of the standard uniform laminas – rectangles, circles, sectors and triangles
Examples include
- two rectangular laminas made into a L-shaped shelf
- a semi-circular lamina cut from a larger semi-circular lamina to create a rainbow shaped sign
- two overlapping circular laminas
In Mechanics 2 both uniform (this Revision Note) and non-uniform laminas (Revision Note 2.1.7) are considered

What modelling assumptions are used with composite laminas?

(For this Revision Note) all laminas making a composite lamina are made from the same uniform material
- uniform means the mass per unit area ( $m$ kg) is equal at every point on a lamina
  - the mass of a lamina is proportional to its area ( $A$ m²)
  - i.e. Mass of a lamina = $A m$ kg
- every lamina being made from the same material means that the mass per unit length will be the same for every lamina
  - i.e. $m$ will be the same for every lamina
  - $m$ ’s cancel in the equation for finding the position of the centre of mass
  - only the value of A(i.e. the area of each lamina) is needed in calculations
A composite lamina 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 laminas is negligible

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

In short, find the area and position of the centre of mass for the individual parts of the composite lamina, then treating these as particles with mass equal to the area, use the equation below (Revision Note 2.1.1) to find the position vector of the centre of mass of the composite lamina

$\sum_{} m_{i} r_{i} = \bar{r} \sum_{} m_{i}$

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

Create your own axes if necessary

Split the composite lamina into two or more standard laminas

e.g.

2-1-3-fig1-step1-eg

STEP 2 Calculate the area and the position of the centre of mass for each standard lamina

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

2-1-3-fig2-step-2-table

(For both squares and rectangles, the position of the centre of mass will be their centres; G₁(2.5, 5.5) and G₂(4.5, 5.5) which are easy to ‘see’, similarly the areas of these shapes are simple to calculate so the table can be completed directly.)

STEP 3 Treat G₁ and G₂ as the positions of particles with masses the area of the standard laminas.

Use $\sum m_{i} r_{i} = \bar{r} \sum m_{i}$ to find the position vector of the centre of mass (G) of the composite lamina.

Remember to give the final answer in the required format.

e.g. $9 (2.5 i + 5.5 j) + 9 (4.5 i + 5.5 j) = 18 (x i + y j)$

$18 (x i + y j) = 63 i + 99 j$

$(x i + y j) = 3.5 i + 5.5 j$

So the coordinates of the centre of mass (G) of the composite lamina are(3.5, 5.5)

Looking back at the diagram this seems a sensible answer.

In the example above
- $x$ and $y$ could’ve been treated separately rather than use vector notation
- the line $y$ = 5.5 is an axis of symmetry of the composite lamina so the $y$ -coordinate of the centre of mass could’ve been written down without any calculations being made
Composite laminas also include cases where a standard lamina has been removed from another standard lamina – the worked example below demonstrates this

Worked example

A uniform lamina in the shape of a triangle has a circle of radius 1 cm and centre (3, 5) removed from it as shown in the diagram below.

2-1-3---we-diagram

Find the coordinates of the centre of mass of the resulting lamina, giving your coordinates to three significant figures.

2-1-3-fig3-compo-we-solution

Examiner Tip

Sketch diagram(s) or add to any given in a question.
If not referenced in a question create a coordinate system of your own, making it clear on your diagrams where your origin is.
For composite laminas that have parts removed, treat the parts removed as having negative area.
You do not have to use vector notation; you can treat each dimension separately. However, do ensure your final answer is in the required format.
Beware! The formula booklet lists the formulae for laminas and frameworks together.

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

Revision Note