Calculate the distance of the centre of mass from AB and ALIH of the uniform lamina.

The diagram shows the lamina shape.

To begin, divide the lamina into smaller sections and calculate the area. In this example, the lamina is best divided into sections A, B and C. Calculate the areas. For example, area A is 5x3=15cm², area B is 4x1=4cm², and area C is 5x3cm². Find the total area by adding the individual areas. Total area = 15+4+15=34cm².

Next, we want to find the centre of mass (C.O.M) from AB. To do this, we can use the formula: ȳM_total = y₁m₁ + y₂m₂ + ….. + y_nm_n. However, as the lamina is uniform and no masses are given, we will use the areas instead of masses (M_total and m). The y values are the distance from AB to the centre of each section and ȳ is the vertical position of the c.o.m from AB. Therefore, we substitute in our values to get 34ȳ = (15x1.5) + (4x5) + (15x8.5), which can be simplified to 34ȳ = 170 and therefore, ȳ = 170/34 = 5cm from AB.

To get the c.o.m from ALIH, we do the same again, except this time we replace ȳ with x̄ as we are now working in the horizontal direction finding distances from ALIH. Once again, substitute in the values e.g. 34x̄ = (15x2.5) + (4x2.5) + (15x2.5) which can be simplified to 34x̄ = 85 and therefore, x̄ = 85/34 = 2.5cm from ALIH.

You will notice that these positions are where the letter B is positioned on the diagram, exactly at the centre of the lamina. This makes sense as the lamina is both uniform and symmetrical in both the y and x planes and so, you would expect the centre of mass to be at the centre. Therefore, we could have answered this question simply by looking at the diagram. However, often, exam questions feature a part where you are expected to find the c.o.m by symmetry, and then a part where you need to calculate the c.o.m. So, it is good to practice both methods.