Stable Equilibrium & Stability

What is called an Equilibrium?

Equilibrium is the condition of a system or body in which its state of rest or motion is not changing with time. It is called stable equilibrium in studies of physics. In case of thermal equilibrium, internal energy of the system doesn’t changing with lapse of time.

In the study of mechanics, if a body is in stable equilibrium, it must satisfy the following equilibrium conditions –

Translation equilibrium condition.
Rotational equilibrium condition.

Therefore, if the linear momentum and angular momentum, both of body particles remain constant with time, then the body is said to be in stable equilibrium condition.

Translation Stable Equilibrium

We know that, a force acting on a body can produce linear acceleration producing translation motion. Hence, translation equilibrium condition states that –

If a rigid body is in translation stable equilibrium, then the resultant of all external forces acting on it must be zero.

Therefore, $\quad \sum \vec {F_{ext}} = 0$

Assume that, a body of mass $( M )$ is moving and the acceleration of its centre of mass is $( \vec {a_{cm}} )$ . Then from Newton’s second law of motion, we have –

$\sum \vec {F_{ext}} = M \ \vec {a_{cm}}$

But, $\quad \vec {a_{cm}} = \left ( \frac {d}{dt} \right ) \ \vec {v_{cm}}$

Therefore, for equilibrium $\quad M \left [ \left ( \frac {d}{dt} \right ) \ \vec {v_{cm}} \right ] = 0$

This means that, either $\quad M = 0 \quad$ or $\quad \left [ \left ( \frac {d}{dt} \right ) \vec {v_{cm}} \right ] = 0$

But, mass of a body $( M )$ never be equal to zero. So, $\quad \left [ \left ( \frac {d}{dt} \right ) \vec {v_{cm}} \right ] = 0$

This concludes that, $\quad ( \vec {v_{cm}} )$ is a constant.

Hence, if a body is in translation equilibrium then it will be either at rest i.e. $( v = 0 )$ or moving with a uniform velocity i.e. $( v_{cm} = \text {Constant} )$ .

Therefore, two different situations are possible for translation equilibrium –

Static equilibrium – If a body is in rest, it will be in translation equilibrium. It is called static equilibrium.
Dynamic equilibrium – If a body is moving with constant velocity along a straight line, it will be in translation equilibrium. It is called dynamic equilibrium.

Rotational Stable Equilibrium

We know that, a torque or moment acting on a body can produce angular acceleration resulting in rotational motion. Hence, rotational equilibrium condition states that –

The resultant of torque and moments about any point due to external forces acting on a body must be zero.

Therefore, $\quad \sum \left ( \vec {\tau_{ext}} \right ) = 0$

TO BE NOTED –

For rotational equilibrium, the choice of point of rotation i.e. the fixed point is not so important. Because if the total torque is zero about any one point then it must be zero about any other point too.

Conditions for Static Equilibrium

Engineering and scientific calculations are based on static equilibrium condition. Thus, detail study of static equilibrium condition is essential.

Static equilibrium states that –

If a system of co-planar forces is in static equilibrium, the sum of resolved parts of forces in any two perpendicular directions must be equal to zero and also the algebraic sum of their moments about any point in their plane must be zero.

A system of many co-planar forces ultimately will reduces to the following –

A resultant force $( R )$
A couple or moment $( M )$

Therefore, general conditions for a system to be in equilibrium are –

Geometrical sum of all the forces must be zero. Therefore, $( R = 0 )$
Geometrical sum of all couples or moments of forces about any point is zero. Therefore, $( M = 0 )$

Any force can be represented into components in two mutually perpendicular directions. This is called resolving forces in rectangular components.

Let, all forces in the system are resolved into components in two mutually perpendicular directions along $XX$ and $YY$ axes.

Then for static equilibrium, following conditions must satisfy –

The vector sum of all component forces in $XX$ direction must be zero. Therefore, $\sum ( {F_x} ) = 0$
The vector sum of all component forces in $YY$ direction must be zero. Therefore, $\sum ( {F_y} ) = 0$
The vector sum of all the moments of forces about any arbitrary point $A$ must be zero. Therefore, $\sum ( {M_A} ) = 0$

TO BE NOTED –

For checking of equilibrium conditions of a system acted by multiple forces, the arbitrary point $A$ ( about which moments are taken ) is so chosen that, most of the forces ( either known force or unknown force ) passes through point $A$ . In this way, moments for most of the forces become zero and so calculations become easier.

See numerical problems based on this article.

Stability of Equilibrium

According to stability, equilibrium of a body is of three types –

Stable equilibrium.
Unstable equilibrium.
Neutral equilibrium.

Stable Equilibrium

If a body tends to regain the equilibrium position after being slightly displaced from equilibrium position and released, then the body is said in stable equilibrium.

In stable equilibrium, following conditions are met –

A body has minimum potential energy.
Centre of mass of body lies below the point of suspension or support.
Centre of mass goes higher when the body is displaced from equilibrium position.
So, potential energy of body increased when displaced from equilibrium position.

EXAMPLE –

A book laying on a horizontal surface.
Bodies laying on floor such as table, chair etc.

Unstable Equilibrium

If a body gets further displaced from its equilibrium position after being slightly displaced and released, then the body is said in unstable equilibrium.

In unstable equilibrium, following conditions are met –

A body has maximum potential energy.
Centre of mass of body lies above the point of suspension or support.
Centre of mass goes lower when it is displaced from equilibrium.
So, potential energy gets reduced when displaced from equilibrium position.

EXAMPLE –

A pencil standing vertically on its tip.
A thin rod standing vertically on a horizontal surface.

Neutral Equilibrium

If a body stays in equilibrium position even after being slightly displaced and released, it is said in neutral equilibrium.

In neutral equilibrium, following conditions are met –

Centre of mass of body lies just at the point of suspension or support.
Centre of mass neither raised nor lowered when body is displaced from equilibrium position.
Potential energy remains constant even if it is displaced from equilibrium.

EXAMPLE –