Torque Characteristics and Formulas, Exercises

The torque, torque or moment of a force is the ability of a force to cause a turn. Etymologically it receives the name of torque as a derivation of the English word torque, from latin torquere (twist).

The torque (with respect to a given point) is the physical magnitude that results from making the vector product between the position vectors of the point where the force is applied and that of the exerted force (in the order indicated). This moment depends on three main elements.

The first of these elements is the magnitude of the applied force, the second is the distance between the point where it is applied and the point with respect to which the body rotates (also called the lever arm), and the third element is the angle of application of said force.

The greater the force, the greater the spin. The same happens with the lever arm: the greater the distance between the point where the force is applied and the point with respect to which it produces the turn, the greater this will be.

Obviously, the torque is of special interest in construction and industry, as well as it is present in countless applications for the home, such as when a nut is tightened with a wrench..

Article index

• 1 Formulas
  • 1.1 Units
• 2 Features
• 3 Resulting torque
• 4 Applications
• 5 Exercises solved
  • 5.1 Exercise 1
  • 5.2 Exercise 2
• 6 References

Formulas

The mathematical expression of the torque of a force with respect to a point O is given by: M = r x F

In this expression r is the vector that joins the point of O with the point P of application of the force, and F is the vector of the applied force.

The units of measurement of the moment are N ∙ m, which although dimensionally equivalent to the Joule (J), have a different meaning and should not be confused.

Therefore, the modulus of the torque takes the value given by the following expression:

M = r ∙ F ∙ sin α

In this expression, α is the angle between the force vector and the vector r or lever arm. The torque is considered positive if the body rotates counterclockwise; on the contrary, it is negative when it rotates clockwise.

Units

As already mentioned above, the unit of measurement of the torque results from the product of a unit of force and a unit of distance. Specifically, the International System of Units uses the newton meter whose symbol is N • m.

At a dimensional level, the newton meter may seem equivalent to the joule; however, in no case should July be used to express moments. The joule is a unit for measuring works or energies that, from a conceptual point of view, are very different from torsional moments.

In the same way, the torsion moment has a vector character, which is both the scalar work and energy.

Characteristics

From what has been seen, it follows that the torque of a force with respect to a point represents the capacity of a force or set of forces to modify the rotation of said body around an axis that passes through the point..

Therefore, the torsional moment generates an angular acceleration on the body and is a magnitude of a vector character (so it is defined from a module, a direction and a sense) that is present in the mechanisms that have been subjected torsion or flexion.

The torque will be zero if the force vector and the vector r have the same direction, since in that case the value of sin α will be zero.

Resulting torque

Given a certain body on which a series of forces acts, if the applied forces act in the same plane, the torque resulting from the application of all these forces; is the sum of the torsional moments resulting from each force. Therefore, it is true that:

M_T = ∑ M = M₁ + M_{two +} M₃ +...

Of course, it is necessary to take into account the sign criterion for torsional moments, as explained above.

Applications

Torque is present in such everyday applications as tightening a nut with a wrench, or opening or closing a faucet or a door.

However, its applications go much further; the torque is also found in the axes of the machinery or in the result of the efforts to which the beams are subjected. Therefore, its applications in industry and mechanics are many and varied.

Solved exercises

Below are a couple of exercises to facilitate understanding of the above.

Exercise 1

Given the following figure in which the distances between point O and points A and B are respectively 10 cm and 20 cm:

a) Calculate the value of the torque modulus with respect to point O if a force of 20 N is applied at point A.

b) Calculate what must be the value of the force applied in B to achieve the same torque as obtained in the previous section.

Solution

In the first place, the data should be transferred to units of the international system.

r_TO = 0.1 m

r_B = 0.2 m

a) To calculate the modulus of the torque we use the following formula:

M = r ∙ F ∙ sin α = 0.1 ∙ 20 ∙ 1 = 2 N ∙ m

b) To determine the requested force proceed in a similar way:

M = r ∙ F ∙ sin α = 0.2 ∙ F ∙ 1 = 2 N ∙ m

Solving for F it is obtained that:

F = 10 N

Exercise 2

A woman exerts a force of 20 N on the end of a 12-inch long wrench. If the angle of the force with the wrench handle is 30 °, what is the torque on the nut??

Solution

The following formula is applied and the operation is carried out:

M = r ∙ F ∙ sin α = 0.3 ∙ 20 ∙ 0.5 = 3 N ∙ m

References

1. Moment of force. (n.d.). On Wikipedia. Retrieved on May 14, 2018, from es.wikipedia.org.
2. Torque (n.d.). In Wikipedia. Retrieved on May 14, 2018, from en.wikipedia.org.
3. Serway, R. A. and Jewett, Jr. J.W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole.
4. Marion, Jerry B. (1996). Classical dynamics of particles and systems. Barcelona: Ed. Reverté.
5. Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics. McGraw-Hill.