Physics Tutorial 6.1 - Centre of Mass. Types of Equilibrium Revision Notes

In these revision notes for Centre of Mass. Types of Equilibrium, we cover the following key points:

• The meaning of centre of mass (gravity)
• How to determine centre of mass in regular and irregular objects
• Why centre of mass is important in science?
• What is equilibrium and which are the factors affecting it?
• How many types of equilibrium are there? Which are they?

Centre of Mass. Types of Equilibrium Revision Notes

Centre of mass (or gravity) is an equilibrium point inside an object, which is very special regarding the stability of the object itself. It is the only point of the object in which there will be no swings, slants, falls, etc., when an object is hanged through or placed on it.

Centre of mass of an object is usually denoted by C. The number of coordinates required to represent the point C depends on the number of dimensions the situation described involves. Thus, for a long and thin bar, there is only one coordinate needed as it is considered as a one-dimensional object. When the object is a kind of thin plate, two coordinates are enough for the centre of mass. When the object is voluminous, there are three coordinates for the centre of mass C needed.

There are three basic methods to determine the centre of mass in objects. They depend on the shape and physical properties of the objects.

1. In regularly shaped and homogenous objects (i.e. in objects made from the same material), centre of mass is determined by geometrical methods.
2. In the irregularly shaped objects, the centre of gravity is determined by hanging them at two different points and then finding the intercept of the two string extensions.
3. When the objects are not homogenous, centre of mass is determined through mathematical methods.

Not all objects are equally stable when they are at rest. Some objects are more stable and they hardly move from their position when a force acts on them. On the other hand, for some other objects it a very small force is enough to make it fall sideways. Based on this criterion, there are 3 types of equilibrium:

1. Stable equilibrium

In this kind of equilibrium, objects are very stable. If a small force acts on them, they shake around but finally they regain the initial position. The condition to have stable equilibrium is

where y_C is the vertical coordinate of the centre of gravity C and h is the height of the object.

2. Unstable equilibrium

Unstable equilibrium is the opposite of stable equilibrium, i.e. a very small force is enough to make an object topple sideways. In other words, a much greater effort is needed to re-establish the equilibrium than to distort it. The condition to have unstable equilibrium is

3. Neutral equilibrium

In this kind of equilibrium when using a force to distort the equilibrium of an object, it turns again at the original position when applying the same force but in opposite direction as before. The condition to have unstable equilibrium is

There are some examples in which objects are leaned but they still don't fall sideways. In other examples, objects start falling sideways despite they apparently seem at vertical position. As long as the vertical line drawn from the centre of gravity falls inside the lower base of the object, it doesn't fall sideways. When the object leans at such an extent that the vertical line drawn from the object's centre of gravity falls outside its lower base, the object falls sideways.

There exists a kind of classification regarding stability even within the same category of equilibrium. It is determined by the position of centre of gravity in respect to the ground. Higher the centre of gravity, less stable the object is.

In summary, the equilibrium or stability of objects depends of three factors:

1. The position of centre of mass in respect to half of an object's height (stable, unstable, neutral),
2. The position of normal line drawn drawn from the centre of mass in respect to the object's base [inside the base - stable (the object stands still), outside the base - unstable (the object falls down)], and
3. The position of centre of gravity in respect to the ground (higher the centre of gravity, less stable the object is).

Some applications of centre of mass in Physics, include:

1. In motion - It helps determining correctly the displacement of an object,
2. In collisions - In head to head collision objects will move in the opposite direction after the contact while if the collision does not occur in the direction of objects' centre of gravity, they will move in different directions after the collision.