Static Equilibrium

An even contest of a tug of war match could contain much effort and energy but very little movement; the force exerted on the rope by both teams may be identical and hence neither team moves an inch. The reason that there is no motion is due to the rope being in static equilibrium.

Static Equilibrium Definition

Static equilibrium is a concept that we observe daily but usually take no notice of. It can be defined as follows.

This sounds complicated but if we think of the evenly contested tug of war example; the force that each team exerts on the rope is equal but in opposite directions so the sum of the forces is zero. There is no turning or rotational motion in the rope so the torque is zero as well. Lastly, the rope and players are not moving, or are at rest. This system meets all of the conditions above so it is said to be in static equilibrium.

The forces causing static equilibrium may be vastly different between systems but the concept remains the same. Static equilibrium can be achieved irrespective of the type of forces acting on the objects in that system.

Static Equilibrium Equation

We will consider two scenarios to illustrate the concept of static equilibrium; the first will be a situation in which there are only forces and no torques and the second will contain both forces and torques.

Scenario 1: Forces Only

We can look at an example to determine an equation that will explain static equilibrium mathematically. As per the figure below, let's assume that a stationary block on a rough floor experiences an applied force \(F\) to the right, with a frictional force \(f\) opposing it, acting to the left. The weight of the box \(W\) acts through its centre and vertically downward and the normal contact force \(N\) acts upward on the block.

Fig. 1. Four forces act in different directions on a block. The opposing forces are equal in magnitude keeping it in static equilibrium since it is not moving,

The normal force and weight are equal in magnitude but opposite in direction to each other, and the same holds for the applied force and frictional force. The effect of friction cancels the effect of the applied force and the effect of the weight cancels the effect of the normal force; the block does not move since the sum of the forces and the sum of the torques on it are both zero and it was initially stationary. The box is said to be in static equilibrium because:

1. the sum of all the forces on the block is zero,

2. no torques are acting on the block, so the total torque is zero, and

3. the block remains at rest.

Mathematically we can write these conditions as follows:

1. \(F-f=0\) and \(N-W=0\),
2. \(T=0\),
3. \(v=0\).

using \(T\) to represent the torque on the block and \(v\) to represent the block's speed. Note that the "minus" signs arise from the fact that\(F\) and \(f\) are in opposite directions, as are \(N\) and \(W\).

Scenario 2: Forces and Torques

Let us now consider a case where torques are involved, and for this, we'll use the example of a uniform see-saw as in the figure below. A boy and girl both stand on opposite sides of the pivot point, which is at the centre of the see-saw's beam. The weight of the boy is \(W_1\) and the weight of the girl is \(W_2\). The normal contact force \(N\) that the pivot exerts on the beam is equal to the sum of the weights of the boy and girl but in the opposite direction (upward). The torque that is exerted on the beam due to the weight of the boy is \(T_1\), the torque due to the girl's weight is \(T_2\), and these torques are counterclockwise and clockwise, respectively. The torques are equal in magnitude so the see-saw does not rotate.

This see-saw does not move and we can say that it is in static equilibrium because the criteria above are all met:

1. the sum of all the forces on the see-saw is zero,

2. the sum of all the torques acting on the see-saw is zero, and

3. the see-saw remains at rest.

We can again write these mathematically which gives us the following equations:

1. \(W_1+W_2-N=0\),
2. \(T_1-T_2=0\),
3. \(v=\omega=0\).

where \(v\) and \(\omega\) represent the linear speed and angular speed of the see-saw respectively.

General Equations

We can use the equations above to write a general mathematical form of the criteria for static equilibrium. Using \(F_i\) to represent the forces acting on a system, \(T_i\) to represent the torques, and \(v\) and \(\omega\) to represent the linear and angular speeds respectively, the general equations can be written as follows:

1. \(F_i=0\),
2. \(T_i=0\),
3. \(v=\omega=0\).

These equations will apply to all systems that are in static equilibrium. If one of the equations is not true for a system, then that system is not in static equilibrium.

Static Equilibrium vs Dynamic Equilibrium

We can now compare static equilibrium and dynamic equilibrium. The two are very similar with just a single difference: in dynamic equilibrium, the system may be moving. It is easy to see that the last criterion is the only difference between the two types of equilibria. We can tabulate the criteria that define static and dynamic equilibria to see this difference.

The last criterion comes from Newton's first law; an object will remain at rest or move with constant speed if the sum of the forces on it is zero. If the object is not moving with constant speed, then the sum of the forces is not zero and the first criterion is violated; the system will neither be in static nor dynamic equilibrium.

We can look at an example of dynamic equilibrium by considering the case in which the block in scenario 1 above is now moving, as in the figure below. The same four forces act on the block but it is moving with a constant speed \(v\) to the right.

Fig. 3. Four forces act in different directions on a block. The opposing forces are equal in magnitude keeping it in dynamic equilibrium since it is moving with constant speed.

Static Equilibrium Example

Let us use scenario 2 above as an example and a chance to test out our knowledge of static equilibrium.

Resultant Force and Static Equilibrium

We want to now see the connection between static equilibrium and resultant force. The resultant force can be defined as follows.

This means that we can rewrite the criteria that define static equilibrium as below:

1. The resultant force on the system is zero,
2. the sum of all the torques on the system is zero, and
3. all particles/objects in the system are at rest.

The only change is that we introduce the term "resultant force" in the first criterion. It is cumbersome to write out "the sum of all the forces" so the term "resultant force" serves us better since it is equivalent and shorter.