Modern Physics

# Twist moment: features 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 torque as a derivation of the English word torque , from the Latin torquere (twist).

Torque (with respect to a given point) is the physical magnitude resulting from the production of the cross 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 at which it is applied and the point at which the body rotates (also called the lever arm) and the third element is the angle of application of said strength.

The greater the force, the greater the rotation. The same applies to the lever arm: the greater the distance between the point where the force is applied and the point in relation to which the curve is produced, the greater it will be.

Of course, torque is of special interest in construction and industry, as well as in many domestic applications, such as when you tighten a nut with a wrench.

## formulas

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

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

The units of measure for the moment are N ∙ m, which although dimensionally equal to July (J), have a different meaning and should not be confused.

Therefore, the torque modulus assumes 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 lever arm vector ro. Torque is considered positive if the body rotates counterclockwise; on the contrary, it is negative when it rotates clockwise.

### Units

As mentioned above, the unit of measure for torque is the product of a unit of force and a unit of distance. Specifically, in the International System of Units the Newton meter whose symbol is N • m is used.

On the dimensional level, Newton’s meter might look like July; However, in no case should July be used to express moments. July is a unit to measure works or energies that, from a conceptual point of view, are very different from torsional moments.

Likewise, torque has a vector character, which is as much scalar work as energy.

## Characteristics

It follows that the moment of twisting a force with respect to a point represents the capacity of a force or set of forces to modify the rotation of said body about an axis passing through the point.

Therefore, the torsional moment generates an angular acceleration in the body and is a magnitude of the vector character (thus, it is defined from a magnitude, a direction and a direction) that is present in the mechanisms that were submitted to torsional or bending.

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

## resulting torque

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

T = Σ M = M 1 + M 2 + H 3 + …

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

## applications

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

However, its applications go far beyond; Torque is also found in the machine shafts or as a result of the stresses to which the beams are subjected. Therefore, its applications in industry and mechanics are many and varied.

## solved exercises

Below are some exercises to help you understand 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 to point A.

b) Calculate the value of the force applied in B to reach the same torque obtained in the previous section.

#### Solution

First of all, it is convenient to pass the data to the units of the international system.

A = 0.1 m

B = 0.2 m

a) To calculate the torque modulus, 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 manner:

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

In offset F, you get the following:

F = 10 N

### Exercise 2

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

#### Solution

The following formula is applied and operated:

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