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Unstable equilibrium: concept and examples

In mechanics, an object is in unstable equilibrium when when it is slightly displaced from its position by a force, it does not return to it. For example, the ball in the following figure, when it is above the mound, is in unstable equilibrium, since the slightest disturbance will make it roll downhill, without the possibility of returning by its own means.

The ball is in unstable equilibrium in the left position, in indifferent equilibrium in the center and in stable equilibrium on the right.
Figure 1. The ball is in unstable equilibrium in the left position, in indifferent equilibrium in the center and in stable equilibrium on the right.

In contrast, when the ball is on a flat surface, as in the center position, it is said to be in neutral or indifferent equilibrium . A disturbing force can be applied, moving it to one side or the other, but if the force disappears, the ball will become stable again.

Finally, on the right, the ball is at the bottom of a concave container. This position is also one of equilibrium, but stable equilibrium . A force that disturbs the ball will only make it oscillate a little around the original position, to end up returning quietly to it.

Causes of loss of balance

Common objects (and people and animals) lose their balance and fall due to the torque caused by weight , the force that the Earth exerts on all objects near its surface. When you have an extended body, the point where the weight acts is called the center of gravity .

The weight can be balanced thanks to a support, such as that provided by a surface, and in this way the object will not move. But even so, it still has the possibility of turning about some point, because in extended objects the balance of forces is not the only factor to keep them still, but also the place where these forces are applied.

Below is a figure with a pencil balanced on its tip, in unstable balance. Any draft of air will cause it to tip over, but in the meantime, the weight and normal supporting force offset each other. In addition, both forces have the same line of action and this passes through the tip of the pencil, ensuring balance.

But if the pencil is tilted just a little bit, as shown to the right, the line of action of the weight stops passing through the tip, which acts as a pivot. Then the weight produces unbalanced torque and the stylus rotates clockwise.

The pencil resting on its tip is in unstable balance, a small disturbance will cause the weight to produce a torque with respect to the tip and the pencil will fall.
Figure 2. The pencil resting on its tip is in unstable balance, a small disturbance will cause the weight to produce a torque with respect to the tip and the pencil will fall.

Factors that guarantee stability

Stable equilibrium is almost always sought, since unstable equilibrium is, as its name implies, quite precarious. Continuing with the example of the pencil, once it falls and comes to rest horizontally on the surface, the new position is much more stable than when it was standing on the tip.

This is due to the fact that, on the one hand, the center of gravity is closer to the surface and, on the other hand, the support surface of the pencil is much larger.

When the support surface is larger, the normal is more likely to be able to counteract the weight, since the surface is precisely what the normal exerts. And if the distance from the center of gravity to the surface is smaller, the lever arm of the weight is smaller, and therefore the torque is also smaller.

In conclusion, the greater the support base of the object, and the closer its center of gravity to the ground, the lower the probability of overturning and the equilibrium tends to be stable. Babies know this and that is why they tend to crawl first before risking standing up.

And if instead of being supported, the body is suspended from a point, the location of the center of gravity also plays a prominent role when establishing balance, as will be seen shortly in the following examples.

Balance examples

Balance in supported bodies

The equilibrium in supported bodies depends, as said, on:

-How close the center of gravity is to the surface.

-The size of the object’s base.

Consider a cone on a flat table. The most stable position without a doubt is with the base of the cone fully supported on the table. This is the stable equilibrium position, since the cone’s center of gravity is on the axis of symmetry and closer to its base than to the tip.

Indifferent equilibrium is achieved by placing the cone lying down and unstable equilibrium corresponds to the cone on its tip, like the pencil, which might not be an easy task, since at the slightest movement the cone tips over.

Stable, indifferent and unstable equilibrium of a cone resting on a table.
Figure 3. Stable, indifferent and unstable equilibrium of a cone resting on a table.

Balance in suspended bodies

It is common to find suspended bodies that hang from at least one point, such as paintings and lamps. When establishing balance, consider the location of the center of gravity and that of the suspension point.

The situation is easy to visualize with the help of a rectangular cardboard sheet or a rule of homogeneous material. Here the center of gravity coincides with the geometric center of the figure, assuming that the mass of the object is evenly distributed.

To place the sheet in unstable balance, it is suspended from a point that is below the center of gravity, it is even enough to hold the sheet between the fingers without tightening too much, to allow freedom of movement.

A small force is sufficient for the blade to immediately rotate one way or the other. The reason for the rotation is the same as in the case of the supported object: the weight exerts an uncompensated torque that facilitates the rotation of the body.

As the sheet rotates, it passes through a position that is of stable equilibrium, in which the suspension point is above the center of gravity. Around this position it oscillates a bit and finally stops.

If a force is applied again, the blade oscillates again but returns again to that position, in which the suspension point and the center of gravity are aligned with the vertical.

Finally, the indifferent balance is checked by passing a pin just through the center of gravity. If the sheet is rotated to be in different positions, it is seen that there will be no major difference between them.

In conclusion, for bodies suspended in unstable equilibrium, the point of suspension is below the center of gravity. And the opposite for stable equilibrium.

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