Equilibrium

Equilibrium Definition

There are two conditions that are required for an object to be in equilibrium:

• No net force is acting on the object.
• No net torque is acting on the object.

So we can provide a basic physical definition of equilibrium as follows:

This means that the motion of objects in equilibrium will not change with time and they will also keep the same amount of energy. Force is a familiar concept but torque may be new to you. Torque is a type of force that tends to cause a rotation. Torque \(\tau\) is given by the equation

\[\tau=Fd\]

where \(F\) is the force perpendicular to the pivot (\(\mathrm{N}\)) and \(d\) is the perpendicular distance to the pivot (\(\mathrm{m}\)). Thus, torque is measured in \(\mathrm{N\,m}\) rather than in \(\mathrm{N}\) like force. The diagram below shows how you can apply a force to a spanner to cause a torque.

Fig. 2: A spanner can be used to apply a torque to another object. Source: via Wikimedia commons, CC0.

Let's study an example that includes both of these quantities, force and torque, to gain a better understanding of equilibrium. Consider a seesaw with two twins sitting at equal distances on either side, as shown below.

Fig. 3: If twins (represented by squares in this diagram though), who weigh the same, sit on either side of a seesaw at equal distances from the center of balance, the system will be in equilibrium.

The downwards force due to gravity (which is the combined weight of the twins and their seesaw) is balanced by the upwards force at the pivot of the seesaw so the net force is zero. If we presume that they both weigh the same, then the torque due to either child will be equal and in opposite directions, so the net torque will be zero. The net force and the net torque on the system are both zero so it is in equilibrium.

Equilibrium Expression

A system is said to be in equilibrium if it has the two following properties:

1. The linear momentum \(p\) of its center of mass is constant.
2. The angular momentum \(L\) about its center of mass, or any other point, is constant.

These two conditions can also be represented by the following expressions:

\( \begin{align} \vec{p}&=\mathrm{constant} \\ \vec{L}&=\mathrm{constant} \end{align} \)

In situations in which the constants in these equations are equal to zero, the system is said to be in static equilibrium. For example, the seesaw in the example above has no translational motion or rotational motion either (from the reference frame in which we are observing it), so it is in static equilibrium. When a system has a constant velocity or a constant angular velocity (or both), it is said to be in dynamic equilibrium. An example of a system in dynamic equilibrium is a car traveling along a road at a constant velocity. In this situation, the driving force is equal to the drag force on the car. Also, the weight of the car is balanced by the reaction force from the road. The net force is zero and the car is in equilibrium even though it is moving.

Equilibrium Formula

Newton's second law, in its linear momentum form, is given by the following equation:

\[\vec{F}_{\mathrm{net}}=\frac{\Delta \vec{p}}{\Delta t}\]

in which \(\vec{F}_{\mathrm{net}}\) is the net force on a system and \( \Delta \) represents a change in the variable that it is next to. If an object is in equilibrium, then the expression above tells us that its linear momentum must be constant. We know that if \(\vec{p}\) is constant then \(\frac{\Delta \vec{p}}{\Delta t}\) is zero and hence the net force must be zero,

\[\vec{F}_{\mathrm{net}}=0\]

and we have arrived back at what we stated at the start - the net force on an object in equilibrium is zero. Similarly for rotational motion, we can relate the net torque on a system to its angular momentum using the following equation:

\[\tau_{\mathrm{net}}=\frac{\Delta L}{\Delta t}\]

The net torque on an object is equal to the rate of change of the object's angular momentum. This is Newton's second law applied to angular momentum. Again, we know that if \(L\) is constant then \(\frac{\Delta L}{\Delta t}\) is zero and so the net torque must be zero.

\[\tau_{\mathrm{net}}=0\]

We can thus state the two requirements for a system to be in equilibrium:

1. The vector sum of all of the forces acting on the body must be zero.
2. The vector sum of all of the external torques acting on the body, measured about any point, must be zero.

We have arrived again at our two conditions for equilibrium that were stated at the start of the article!

We can also use the requirement for equilibrium that the net torque must be zero to find unknown quantities for systems in equilibrium. Consider again the seesaw from above. Imagine that one of the twins was replaced by their older brother, who happens to weigh twice as much. He sits at a distance from the center of the seesaw so that it remains balanced. How could we find this distance? We know the equation for torque to be

\[\tau=Fd\]

The force has doubled due to the weight of the older brother being double which means that he must sit at half the distance for the torque to be the same as before!

Stable Equilibrium

You might have heard of a stable equilibrium before, but make sure not to get it confused with static equilibrium! Systems in stable equilibrium have the property that if they are displaced a small amount from their static equilibrium position by a force, they will return to this state of static equilibrium after the force has subsided.

You can probably guess what will happen to a system in unstable equilibrium. If a system in unstable equilibrium is displaced a small amount by a force, the object will no longer be in equilibrium when the force is removed.

Equilibrium Examples

The conditions for equilibrium above can be used to simplify many situations and solve many problems in terms of simple equations.