TRANSLATIONAL EQUILIBRIUM: DETERMINATION, APPLICATIONS, EXAMPLES - PHYSICAL - 2025

The translational equilibrium is a state in which an object as a whole is when all forces acting thereon are offset, giving as a result a net force zero. Mathematically it is equivalent to saying that F ₁ + F ₂ + F ₃ +…. = 0, where F ₁, F ₂, F ₃ … are the forces involved.

The fact that a body is in translational equilibrium does not mean that it is necessarily at rest. This is a particular case of the definition given above. The object may be in motion, but in the absence of acceleration, this will be a uniform rectilinear motion.

Figure 1. Translational balance is important for a large number of sports. Source: Pixabay.

So if the body is at rest, it continues like this. And if it already has movement, it will have constant speed. In general, the motion of any object is a composition of translations and rotations. The translations can be as shown in figure 2: linear or curvilinear.

But if one of the object's points is fixed, then the only chance it has to move is to rotate. An example of this is a CD, whose center is fixed. The CD has the ability to rotate around an axis that passes through that point, but not to translate.

When objects have fixed points or are supported on surfaces, we speak of links. The links interact by limiting the movements that the object is capable of making.

Determination of translational equilibrium

For a particle in equilibrium it is valid to ensure that:

F _R = 0

Or in summation notation:

It is clear that for a body to be in translational equilibrium, the forces acting on it must be compensated in some way, so that their resultant is zero.

In this way the object will not experience acceleration and all its particles are at rest or undergoing rectilinear translations with constant speed.

Now if objects can rotate, they generally will. That is why most movements consist of combinations of translation and rotation.

Rotating an object

When rotational balance is important, it may be necessary to ensure that the object does not rotate. So you have to study if there are torques or moments acting on it.

Torque is the vector magnitude on which the rotations depend. It requires a force to be applied, but the point of application of the force is also important. To clarify the idea, consider an extended object on which a force F acts and let us see if it is capable of producing a rotation about some axis O.

It is already intuited that by pushing the object at point P with the force F, it is possible to make it rotate around point O, with a counterclockwise rotation. But the direction in which the force is applied is also important. For example, the force applied to the figure in the middle will not make the object rotate, although it can certainly move it.

Figure 2. Various ways of applying a force on a large object, only in the figure on the extreme left a rotation effect is obtained. Source: self made.

Applying force directly to point O will not turn the object either. So it is clear that to achieve a rotational effect, the force must be applied at a certain distance from the axis of rotation and its line of action must not pass through that axis.

Definition of torque

The torque or moment of a force, denoted as τ, the vector magnitude in charge of putting all these facts together, is defined as:

The vector r is directed from the axis of rotation to the point of application of the force and the participation of the angle between r and F is important. Therefore, the magnitude of the torque is expressed as:

The most effective torque occurs when r and F are perpendicular.

Now, if it is desired that there are no rotations or these take place with constant angular acceleration, it is necessary that the sum of the torques acting on the object is zero, analogously to what was considered for the forces:

Equilibrium conditions

Balance means stability, harmony and balance. For the movement of an object to have these characteristics, the conditions described in the previous sections must be applied:

1) F ₁ + F ₂ + F ₃ +…. = 0

2) τ ₁ + τ ₂ + τ ₃ +…. = 0

The first condition guarantees translational equilibrium and the second, rotational equilibrium. Both must be fulfilled if the object is to remain in static equilibrium (absence of movement of any kind).

Applications

Equilibrium conditions are applicable to many structures, since when buildings or diverse objects are built, it is done with the intention that their parts remain in the same relative positions with each other. In other words, the object does not come apart.

This is important for example when building bridges that stay firmly underfoot, or when designing habitable structures that do not change position or have a tendency to tip over.

Although it is believed that uniform rectilinear motion is an extreme simplification of motion, which rarely occurs in nature, it must be remembered that the speed of light in vacuum is constant, and that of sound in air, too, if consider the medium homogeneous.

In many man-made mobile structures it is important to maintain a constant speed: for example, on escalators and assembly lines.

Examples

This is the classic exercise of the tensions that hold the lamp in balance. The lamp is known to weigh 15 kg. Find the magnitudes of the stresses necessary to hold it in this position.

Figure 3. The equilibrium of the lamp is guaranteed by applying the translational equilibrium condition. Source: self made.

Solution

To solve it, we focus on the knot where the three strings meet. The respective free-body diagrams for the node and for the lamp are shown in the figure above.

The weight of the lamp is W = 5 Kg. 9.8 m / s ² = 49 N. For the lamp to be in equilibrium, it is sufficient for the first equilibrium condition to be fulfilled:

The voltages T ₁ and T ₂ must be decomposed:

It is a system of two equations with two unknowns, whose answer is: T ₁ = 24.5 N and T ₂ = 42.4 N.

References

1. Rex, A. 2011. Fundamentals of Physics. Pearson. 76 - 90.
2. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7 ^ma. Ed. Cengage Learning. 120-124.
3. Serway, R., Vulle, C. 2011. Fundamentals of Physics. 9 ^na Ed. Cengage Learning. 99-112.
4. Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. MacGraw Hill. 71 - 87.
5. Walker, J. 2010. Physics. Addison Wesley. 332 -346.