Centres of Mass of Plane Figures

Understanding the Concept of Centre of Mass

The centre of mass of a plane figure is the point at which the whole mass of the figure may be assumed to be concentrated.
A composite plane figure is an object that can be separated into simpler shapes such as rectangles, circles, or triangles, for which the properties and centres of mass are known.

Calculating Centre of Mass For Regular Shapes

The centre of mass of a regular geometric shape, such as a square, rectangle, or circle, lies at the geometric centre of the shape.
Centre of mass for a triangle lies at the point where its medians intersect, which is two-thirds of the distance from any vertex to the midpoint of the opposite side.
For a uniform semi-circle, the centre of mass is located along the line of symmetry at a distance of 4R/3π from the flat side.

Calculating Centre of Mass For Irregular and Composite Shapes

To find the centre of mass for a composite plane figure, you start by separating it into simpler shapes and find the centre of mass of each individual shape.
Once each individual centre of mass has been found, they can be used to calculate the overall centre of mass using formulae for the x and y coordinates: X = (m1x1 + m2x2 + m3x3 +…)/(m1 + m2 + m3 +…) , Y = (m1y1 + m2y2 + m3y3 +…)/(m1 + m2 + m3 +…).
Keep in mind that in these formulas, m refers to the mass of the individual shape and x or y refers to the coordinates of the individual shapes’ centre of mass.

Applying the Principle of Moments

The principle of moments states that for a body to be in equilibrium, the sum of the clockwise moments about any point must equal the sum of the anticlockwise moments about that same point.
Use the principle of moments to calculate the unknown mass or distance in a system of particles or a rigid body.

Understanding the Impact of Varying Mass Distribution

When a body’s mass distribution is changed, its centre of mass also changes. If a mass is added or removed, the centre of mass shifts towards or away from the added or removed mass.
Applying a force away from a body’s centre of mass creates a turning effect, which causes the body to rotate.