Centre of Mass. Types of Equilibrium

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In this Physics tutorial, you will learn:

  • 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?

Introduction

Can you make an egg stay at rest when you place it endwise on a table (as Christopher Columbus once challenged the participants in a dinner when they tried to depreciate his achievement of discovering America)? Why?

What happens when you hang a stick on a string? Does the stick stay horizontally? Why?

What is Centre of Mass?

Suppose you hang a rectangle made of cardboard at the middle of the longest side, as the one shown in the figure.

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When you release it, the cardboard may start to swing as shown below.

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However, after a while, it eventually stops swinging, at the original position shown in the first figure. This is because the right half of the rectangle is equally heavy as the left half, i.e. these is a certain symmetry in the distribution of gravitational force acting on both halves of the cardboard rectangle. The string here acts as a line of symmetry between the two halves of the rectangle if we virtually extend it so that the extension passes through the rectangle as show below.

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But, when you cut a piece (let's say one quarter) from the rectangle as shown in the figure below,

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it is impossible for the object to stay in this position after released. It will swing and occupy a final (equilibrium) position like the one shown below.

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This is because there is no symmetry anymore; the right part of the figure is twice as heavier as its left part, and therefore, the Earth attracts it more. As a result, the right part of the cardboard figure will lower more than its left part in order to achieve stability.

It is clear that the imaginary extension of the string passing through the cardboard object cannot literally be considered anymore as a line of symmetry, as it is obvious that the right piece is not identical to the left one. However, we can think it as a line that divides the cardboard object in two parts of equal weight as the object tries to occupy such a position in which it responds evenly to the gravitational attraction. Look at the figure.

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The yellow-coloured part has the same area as the rose-coloured one. As a result, it has the same weight as well. Therefore, we can think the extension in the direction of the string, (i.e. in the vertical direction) that passes through the hanged object as a "line of equilibrium."

If we hang the same object at another position, obviously, there will be another "line of equilibrium" (determined by the gravitational attraction) to be drawn, as shown in the figure below.

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From Geometry, it is known that two lines cross at a single point. Therefore, when two lines of equilibrium cross, we obtain a single point, known as equilibrium point, which is very special regarding the stability of the object. In Physics, this point is known as the "centre of mass" or "centre of gravity" and it is the only point of the object in which there will be no swings, slants, falls, etc. when the object is hanged through or placed on it. This means that when the object shown in the figure above is stuck on the wall at the centre of mass using a nail, the object does not swing around to achieve the equilibrium dictated by the gravity as it is already in equilibrium.

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From all said above, it is obvious that we need to hang the object at least in two different positions in order to find its centre of mass (gravity).

Usually, the centre of mass of an object is 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.

How to Determine the Centre of Mass in Objects?

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. Thus, from geometry, it is known that there are seven types of regular objects. They are:

a Cube

Its centre of gravity is at the intersection point of diagonals as shown below:

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b. Cuboid

Its centre of gravity is determined by using the same method as for the cube.

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c. Prism

For illustration, let's take a regular triangular prism. Its centre of mass is at the midpoint of the segment passing through the median intercepts of the two bases.

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d. Pyramid

Its centre of mass is at the segment that connects the upper vertex and the base centre. This segment represents the height of pyramid. Thus, centre of mass is at 1/3 of height starting from the ground.

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e. Cylinder

Again, like in prisms, centre of mass in a cylinder is the midpoint of the segment that connects the centres of the two bases.

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f. Cone

Its centre of mass is determined in a similar way as for pyramid, i.e. at 1/3 of the height starting from the ground.

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g. Sphere

The centre of gravity of a sphere is at its geometrical centre.

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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 as discussed earlier.

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3. When the objects are not homogenous, mathematical methods are used to determine the centre of mass. We will discuss more extensively about these methods in the next tutorial.

Types of Equilibrium

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. In this paragraph, all possible types of equilibrium and their features are given.

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. Below, two examples of stable equilibrium are shown.

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The cone will shake around for a while when a small force F acts laterally on it, but eventually it will stop at the original position. If we want to make it fall sideways, we must apply a large force.

Another example of stable equilibrium is a ball moving inside a half-spherical hollow as shown below.

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The common feature of these two examples is that they both have the centre of gravity in the lower half of the system. Indeed, the cone has the centre of gravity at 1/3 of its height as stated earlier. Also, it is obvious that the centre of gravity of the small sphere is much below than half the radius of the half-sphere. Look at the figure:

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Hence, if we take the vertical axis (the y-axis) that starts from zero at ground level and goes upwards as reference, the condition to have stable equilibrium is

yC < 1/2 h

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. Look at the figure:

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From the figure, it is clear that centre of mass is above half of the object's height. Therefore, the condition to have unstable equilibrium is

yC > 1/2 h

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. Geometrically, centre of mass is at the same level as half the height of the object, i.e. centre of mass is at middle of the object. Look at the figure:

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Sphere and cube are the two most outstanding examples of neutral equilibrium. In both of them,

yC = 1/2 h

Example 1

Find the type of equilibrium for the objects shown below.

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Solution 1

a.The ball is in unstable equilibrium as its centre of gravity is above the half-sphere, i.e. much higher than half radius as shown below.

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b.The object shown in the figure is in stable equilibrium as since it is wider in the lower part, its centre of mass is below half of height.

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c.This object is in unstable equilibrium since it is wider in the upper part, so its centre of mass is above half of height.

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d.This object is in neutral equilibrium as its centre of mass is at the level of half-height.

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When Does an Object Start Falling Sideways?

There are some examples in which objects are leaned but they still don't fall sideways (Leaning Tower of Pisa for instance). 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. Look at the figure:

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More Stable and Less Stable within Neutral

Consider a cuboid or a cylinder. They both represent examples of neutral equilibrium in all positions because centre of gravity is at the same level as half of object's height. However, there is a big difference regarding the equilibrium of these object dictated by the position in which they are placed on the ground, although in all cases this equilibrium is neutral. Look at the figures below.

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In both figures the equilibrium is neutral because the centre of gravity is at halh of the height. However, if the object is placed as shown in the first figure, it is more stable than if it is placed as shown in the second figure because in the first case, the centre of gravity is nearer to the ground. Therefore, there exists a kind of classification regarding stability even within the same category of equilibrium.

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).

The Importance of Centre of Mass in Physics

There are a lot of applications of centre of mass concept in physics. Below are mentioned a few of them.

a. In motion

If a moving object combines both linear and circular motion at the same time, we obtain different results for displacement depending on where we start measuring the values. Look at the figure in which a bus is shown from above.

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It is not difficult to observe that if we measure the displacement starting from the front part of the bus (path 1), we will get a smaller value than if displacement is measured starting from rear of the bus (path 2). Furthermore, the direction is also different. Therefore, to avoid such problems, we must calculate the displacement of an object considering the initial and final position of its centre of mass as shown below.

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Hence, the arrow shown by Δr, whose tip and tail are both at the centre of mass of the object, represents the correct displacement of the bus.

b. In collisions

A head to head collision occurs when two objects collide in the direction of their respective centres of gravity. As a result, after the collision, objects will move in the opposite direction as before as shown below.

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But if the collision does not occur in the direction of objects' centre of gravity, they will move in different directions after the collision, as shown below.

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Summary

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

yC < 1/2 h

where yC 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

yC > 1/2 h

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

yC = 1/2 h

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.

Centre of Mass. Types of Equilibrium Revision Questions

1) Four objects are hanged on strings as shown in the figure and initially they are not allowed to move.

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Which object will move after being released?

  1. Object A
  2. Object B
  3. Object C
  4. Object D

Correct Answer: D

2) What are the coordinates of the centre of mass of the object shown in the figure below if xO = 6 units and zO = 21 units? Write the coordinates of centre of mass in the format (xC, yC, zC).

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  1. (6, 6, 21)
  2. (6, 0, 7)
  3. (6, 6, 7)
  4. (6, 6, 14)

Correct Answer: C

3) Four objects, A, B, C and D are at rest on the same horizontal plane as shown below.

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A strong earthquake occurs and the objects start topple over one by one. Which is the order of their toppling occurrence starting from the earliest?

  1. D - C - A - B
  2. B - C - A - D
  3. D - A - C - B
  4. C - D - A - B

Correct Answer: C

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