An object is said to be in equilibrium if the net forces and net torques are zero.
A typical case of equilibrium is of an object immersed in the gravitational field. In such case, the weight is a force always present. It can be demonstrated that in situations where the weight of a body has to be taken into account, the net force of the gravitational forces on all particles of the body can be considered as a single weight acting vertically downwards through the centre of gravity of the body.
The centre of gravity of geometrically shaped objects (of homogeneous density) will lie in the geometric centre. The centre of gravity of different geometrically shaped objects are shown in figure below.
| Object | centre of gravity | |
| a. | Uniform straight wire | the mid point |
| b. | Uniform beam or rod | the mid point of the axis |
| c. | Circle | the center of the circle |
| d. | Ring | its center |
| e. | Triangle | Centroid |
| f. | rectangle | intersecton of its diagonals |
| g. | rhombus | intersecton of its diagonals |
| h. | Sphere | its center |
| i. | Solid cone | 1/4th of its height from its base |
| j. | hollow cone | 1/3rd of its height from its base |
| k. | cube/cuboid | the mid point of the line joining the centres of opposite sides |
A practical procedure to determine the centre of gravity of an irregularly shaped lamina using a plumb line is described below.
A typical case is of an object immersed in the gravitational field. In equilibrium, the torque created by the weight is zero, because its line of action passes through the point of resting. It the body of the object is pushed slightly, the line of action of the weight stops passing through the point of pivot and the reaction force creates a couple that rotates the body.
If the initial rotations tends to restore the original position, the equilibrium is said to be stable. But if the rotation generated by the weight couple rotate the object to a position where the couple increases, the equilibrium is said to be unstable. This happens when the body is tilted so the vertical line drawn from the centre of gravity (c.g) falls outside the base.
Due to the fact that the weight is always a downwards force, its torque created together with the reaction on a point of the base, tends to keep the centre of gravity of the body as lower as possible.
A body is said to be in stable equilibrium on a surface when the centre of gravity goes up when a parallel-to-the-surface force is applied or when it is displaced from its position. This happens when a vertical line drawn from the centre of gravity (line action of the weight) falls within the base of support. The stable equilibrium is the condition of stability.
A body is said to be in unstable equilibrium on a surface when the centre of gravity goes down when a parallel-to-the-surface force is applied or when it is displaced from its position.
The body is said to be in neutral equilibrium when the center of gravity of a body neither goes up nor goes down when it is displaced from its position. Take the example of a ball or a cylinder rolling on the ground. Even though the body is moving, the height of centre of gravity from the ground level remains unchanged.
An object is said to be stable (or keeps a stable equilibrium) if it is steady and well balanced so that when it is pushed slightly it does not topple or fall off easily.
Keeping all the rest of conditions equal,
When a ship floats in the water the forces of buoyancy and gravity balance each other because they are equal.
The following three diagrams show how loads affect the centre of gravity and stability of a ship. A fully loaded ship [figure (a)] brings the centre of gravity and the centre of buoyant force close together making the ship stable.
When the ship is unloaded [figure (b) above] the centre of the gravity and the centre of buoyancy have moved far apart, then the ship will be unstable.
In the figure (c) above, weight of the flooded ballast tanks restore balance.
A tight rope walker in a circus carries a weighted pole or an umbrella.
These are the toys called 'mobiles' which look unstable and yet do not fall (figure shown below). The weights of these toys are so adjusted that their centre of gravity is very much near the base. As a result, any push to the toy tends to raise the center of gravity. When the force is removed, the mobiles swing back and forth about its stable rest position.
Conditions for the rigid body to be at equilibrium
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The figure shows dishes kept at different orientation, on each of
which a marble is placed. If the marble in figure (a) is displaced
slightly, it would rattle about for a while and come to rest in its
original place.
This is an example of stable equilibrium. A small disturbance to a system in such a state does not produce any drastic change and the system eventually returns to its original configuration. On the other hand, if the delicately poised marble in figure (b) is displaced even slightly it would roll over and not get back. This is an example of unstable equilibrium.
The marble in figure (c) will roll over only if given a push. This is an example of metastable equilibrium. The marble on the flat plate is equally comfortable wherever it is and is said to be in neutral equilibrium.
