Equilibrium of the Body: Classification, Examples

Equilibrium of the Body: Equilibrium in physics is the system’s condition when neither its state of motion nor its internal energy changes with time. When a simple mechanical body does not experience linear acceleration and angular acceleration, it is known to be in equilibrium. Unless it’s disturbed by an external force, it will maintain its particular condition indefinitely. For a point mass, the equilibrium arises when the resultant of all the forces acting upon the particle is zero. But for a rigid body to be in equilibrium, the body should have both rotational and translational equilibrium. For this, the vector sum of all the torques as well as force should be zero. There are different types of equilibriums. In this article, we will learn more about equilibrium and types of equilibrium with examples.

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Types of Equilibrium of the Body

With respect to the state of a body (moving or at rest), we can divide the equilibrium into two categories. These are as follows,
1. Dynamic equilibrium.
2. Static equilibrium.

Dynamic equilibrium:

When a body is in a state of uniform motion, and the resultant total force and resultant torque are zero, it is said to be in dynamic equilibrium.
Examples:
(i) Motion of a ball on a frictionless floor.
(ii) Rotation of the earth about its axis.

Static equilibrium:

Static equilibrium is the body’s physical state in which the total force and torque on the body are zero, and the body is in the state of rest. In other words, we can say that static equilibrium occurs when all the forces and torques acting on the object are balanced, and the body is not in motion.
Examples:-
(i) Book resting on a table.
(ii) A car parked on the road.
Based on the system’s potential energy and equilibrium conditions, it can be further divided into three categories. These are,

1. Stable equilibrium
2. Unstable equilibrium
3. Neutral equilibrium

Stable equilibrium:- A body is said to be in stable equilibrium whenever we displace the body by a small distance; the state produces forces that tend to oppose the displacement and returns the body to its state of equilibrium. In this equilibrium, the body will be very stable. The restoring force always tries to bring the system to its original state (stable state). The potential energy of the body will be less in this case. To disturb the body in stable equilibrium, we will need to displace the body from its position by a large amount.
Examples:
1. A ball is placed in the valley between two hills; a small displacement of the ball toward any side will result in a force returning the ball to the first position. Thus, the equilibrium is known as stable.

2. A cone with a flat surface is placed on the horizontal floor. If the cone is slightly tilted, it will come back to its original position.

Learn About Stable And Unstable Equilibrium

Unstable equilibrium: An equilibrium is known as unstable equilibrium whenever we displace the body by a small distance, then the state produces forces that do not tend to oppose the displacement, and it will move away from the body from its original position.
Examples:
1. A cone with a vertex is placed on the horizontal floor on the vertex. If the cone is slightly tilted, it will not come back to its original position.
2. A ball is placed on the top of a rounded hill. Small displacement of the ball toward any side will result in a force that moves the ball away from the first position.

Neutral equilibrium: If a ball is pushed slightly to roll, it will neither come back to its original position nor move forward. It will come to rest after some time.
Examples:
1. A ball is lying on a flat surface. Small displacement will not have any impact on the ball.

2. A cone with an inclined surface is placed on the horizontal floor. If the cone is slightly moved or displaced, it will not come back to its original position, or it will not have any impact on the cone.

Equilibrium and Potential Energy

Equilibrium can be defined in terms of potential energy if only conservative forces are acting on the system. The stability of any system depends on the potential energy of the system with respect to mean position.
For straight-line motion \((1\;\rm{D})\), the change in potential energy \((\rm{dU})\) is given as,
\({\text{d}}U = – {F_x}{\text{d}}x\)
Now for equilibrium, we know that the sum of all the forces should be zero.
Then,
\({F_x} = – \frac{{dU}}{{{\text{d}}x}} = 0\)
For stable equilibrium, the potential energy should be minimum. So, the second derivative should be positive. In this case, for slight displacement, the restoring force should act on the body to bring it to its initial stable position.
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} > 0\)
For unstable equilibrium, the second derivative should be Negative. So, for slight displacement, the force should take away the body from the equilibrium position.
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} < 0\)
For Neutral equilibrium, the second derivative should be zero. So, for slight displacement, there is no force.
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} = 0\).

Summary

Equilibrium in physics is the condition of the system when neither its state of motion nor its internal energy changes with time. We can get the state of equilibrium by just getting the direction of force acting on the body when slightly displaced from the original position. If the force is restoring, i.e., the direction of force is towards the equilibrium position, then it is a stable equilibrium. If the force is away, i.e., the direction of the force is away from the equilibrium position, then it is an unstable equilibrium. If no force acting on the body, then neutral equilibrium. For stable equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} > 0\), for unstable equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} < 0\), and natural equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} = 0\).

Solved Examples on the Equilibrium of the Body

Q.1. The potential energy function of the object confined to move along \(x-\)axis \(x > 0\) is given below:
\(U\left( x \right) = {U_0}\left[ {{{\left( {\frac{k}{x}} \right)}^{12}} – 2{{\left( {\frac{k}{x}} \right)}^6}} \right]\). Here \(k > 0\) and \({U_0} > 0\)
Find the equilibrium position and find out if it is stable, neutral and unstable equilibrium?
Ans: The given equation of potential energy,
\(U\left( x \right) = {U_0}\left[ {{{\left( {\frac{k}{x}} \right)}^{12}} – 2{{\left( {\frac{k}{x}} \right)}^6}} \right]\)
Differentiating with respect to \(x\), we get:
\(\frac{{{\text{d}}U\left( x \right)}}{{{\text{d}}x}} = {U_0}\left[ { – 12\frac{{{k^{12}}}}{{{x^{13}}}} + 12\frac{{{k^6}}}{{{x^7}}}} \right]\)
Now for equilibrium position, the net force on the body must be zero.
\(F = – \frac{{{\text{d}}U\left( x \right)}}{{{\text{d}}x}} = 0\)
\( \Rightarrow {U_0}\left[ { – 12\frac{{{k^{12}}}}{{{x^{13}}}} + 12\frac{{{k^6}}}{{{x^7}}}} \right] = 0\)
\( \Rightarrow x = k\)
Again, differentiating with respect to \(x\), we get:
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} = {U_0}\left[ { + 156\frac{{{k^{12}}}}{{{x^{14}}}} – 84\frac{{{k^6}}}{{{x^8}}}} \right]\)
Now for \(x = k\)
We get:
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} = 72\frac{{{U_0}}}{{{k^2}}}\)
Which is positive as \(k > 0\) and \(U_0 > 0\)
So it is the case of Stable equilibrium.

Q.2. The potential energy \(U\) of an object of unit mass in a one-dimensional conservative force field is given by,
\(U = {x^2} – 4x + 3\)
Find out the equilibrium position if x is in meter.
Ans: The given function of potential energy,
\(U = {x^2} – 4x + 3\)
Now the force acting on the object will be given by,
\(F = – \frac{{{\text{d}}U}}{{{\text{d}}x}} = – \left( {2x – 4} \right)\)
Now for equilibrium position, the net force on the body must be zero.
\( – \left( {2x – 4} \right) = 0\)
\(x = 2\;\rm{m}\)
So \(x = 2\;\rm{m}\) is the equilibrium position.

PRACTICE QUESTIONS RELATED TO EQUILIBRIUM

FAQs on Equilibrium of the Body

Q.1. What is the stable equilibrium of a body?
Ans: A body is said to be in stable equilibrium whenever we displace the body by a small distance, then the state produces forces that tend to oppose the displacement and returns the body to its state of equilibrium. For example, consider a marble at the bottom of a bowl. It will experience a restoring force when it is displaced from its original equilibrium position. This force will always try to move it back towards its equilibrium position.

Q.2. What is the relation between potential energy and states of equilibrium?
Ans: Equation of the different states of equilibrium is given below,
For stable equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} > 0\), for unstable equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} < 0\), and for natural equilibrium \(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} = 0\).

Q.3. What are the types of equilibrium?
Ans: There are three different states of equilibrium which are as follows:
A. Stable equilibrium
B. Unstable equilibrium
C. A neutral equilibrium

Q.4. Which is always true for a body in equilibrium?
Ans: Equilibrium in physics is the condition of the system when neither its state of motion nor its internal energy changes with time. For a body to be in equilibrium, the net force and net torque acting on the body should be zero.

Q.5. Write down the condition to find the point of stable equilibrium.
Ans: For stable equilibrium, the force acting on the body should be zero.
\(F = \frac{{{\text{d}}U\left( x \right)}}{{{\text{d}}x}} = 0\)
Also, the double derivative of the potential function should be positive.
\(\frac{{{d^2}U}}{{{\text{d}}{x^2}}} > 0\)
Where \(U(x)\) is the potential energy function of \(x\).

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