Rotation is a typical type of mechanical movement that is often found in nature and technology. Any rotation arises as a result of the action of some external force on the system under consideration. This force creates the so-called torque. What it is, what it depends on, is considered in the article.
Rotation process
Before considering the concept of torque, we give a description of the systems to which this concept can be applied. The rotation system assumes the presence of an axis in it around which circular motion or rotation is carried out. The distance from this axis to the material points of the system is called the radius of rotation.
From the point of view of kinematics, the process is characterized by three angular quantities:
- rotation angle θ (measured in radians);
- angular velocity ω (measured in radians per second);
- acceleration angular α (measured in radians per second square).
These quantities are related to each other by the following equalities:
ω = dθ / dt;
α = dω / dt.
Examples of rotation in nature are the movements of planets in their orbits and around their axes, the movements of tornadoes. In everyday life and technology, the movement in question is characteristic of engines of engines, wrenches, construction cranes, opening doors and so on.
Determination of the moment of force
Now we turn to the immediate topic of the article. According to the physical definition, the moment of force is a vector product of the vector of application of force relative to the axis of rotation by the vector of the force itself. The corresponding mathematical expression can be written as follows:
M¯ = [r¯ * F¯].
Here, the vector r¯ is directed from the axis of rotation to the point of application of the force F¯.
In this formula of torque M¯, the force F¯ can be directed arbitrarily with respect to the direction of the axis. However, the force component parallel to the axis will not create rotation if the axis is rigidly fixed. In most physics problems, it is necessary to consider the forces F¯ that lie in the planes perpendicular to the axis of rotation. In these cases, the absolute value of the torque can be determined by the following formula:
| M¯ | = | r¯ | * | F¯ | * sin (β).
Where β is the angle between the vectors r¯ and F¯.
What is leverage?
The lever of force plays an important role in determining the magnitude of the moment of force. To understand what is at stake, consider the following figure.
Shown here is a rod of length L, which is fixed at the point of rotation by one of its ends. The force F directed at an acute angle φ acts at the other end. According to the definition of the moment of force, we can write:
M = F * L * sin (180 o -φ).
The angle (180 o -φ) appeared because the vector L¯ is directed from the fixed end to the free. Given the periodicity of the trigonometric function of the sine, we can rewrite this equality in the following form:
M = F * L * sin (φ).
Now let's pay attention to a right-angled triangle built on the sides L, d and F. By definition of the sine function, the product of the hypotenuse L by the sine of the angle φ gives the value of the leg d. Then we come to the equality:
M = F * d.
The linear quantity d is called the lever of force. It is equal to the distance from the force vector F¯ to the axis of rotation. As can be seen from the formula, it is convenient to use the concept of the lever of force when calculating the moment M. The obtained formula indicates that the maximum torque for a certain force F will occur only when the length of the radius vector r¯ (L¯ in the figure above) is equal to leverage, that is, r¯ and F¯ will be mutually perpendicular.
The direction of action of the quantity M¯
It was shown above that torque is a vector characteristic for a given system. Where is this vector directed? It is not difficult to answer this question if we recall that the result of the product of two vectors is a third vector that lies on an axis perpendicular to the plane of arrangement of the original vectors.
It remains to be decided whether the moment of force will be directed up or down (to or from the reader) relative to the mentioned plane. This can be determined either by the rule of the gimlet, or by using the rule of the right hand. Here are both rules:
- Right hand rule. If you position the right hand so that its four fingers move from the beginning of the vector r¯ to its end, and then from the beginning of the vector F¯ to its end, then the thumb protruded will indicate the direction of the moment M¯.
- Rule of the gimlet. If the direction of rotation of the imaginary gimlet coincides with the direction of the rotational motion of the system, then the translational motion of the gimlet will indicate the direction of the vector M¯. Recall that it rotates only clockwise.
Both rules are equal, so everyone can use the one that is more convenient for him.
In solving practical problems, different directions of torque (up - down, left - right) are taken into account using the signs "+" or "-". It should be remembered that the positive direction of the moment M¯ is considered to be one that leads to the rotation of the system counterclockwise. Accordingly, if some force causes the system to rotate along the clock, then its moment will have a negative value.
The physical meaning of M¯
In physics and mechanics of rotation, the quantity M¯ determines the ability of a force or the sum of forces to rotate. Since in the mathematical definition of M¯ is not only a force, but also the radius vector of its application, it is the latter that largely determines the marked rotational ability. To make it clearer what kind of ability we are talking about, here are a few examples:
- Each person, at least once in his life, tried to open the door, not holding the handle, but pushing it close to the hinges. In the latter case, you have to make a significant effort to achieve the desired result.
- To unscrew the nut from the bolt, use special wrenches. The longer the key, the easier it is to unscrew the nut.
- To feel the importance of the lever of power, we suggest readers to do the following experiment: take a chair and try to hold it with one hand on the weight, in one case lean your hand against the body, in the other - perform the task on a straight hand. The latter will be an impossible task for many, although the weight of the chair remains the same.
Units of force moment
A few words should also be said about in which units in SI the torque is measured. According to the formula written down for him, it is measured in Newtons per meter (N * m). However, these units also measure work and energy in physics (1 N * m = 1 joule). The Joule for the moment M¯ is not applied, since the work is a scalar quantity, while M¯ is a vector.
Nevertheless, the coincidence of the units of the moment of force with the units of energy is not accidental. Work on the rotation of the system, perfect moment M, is calculated by the formula:
A = M * θ.
Whence we get that M can also be expressed in joules per radian (J / rad).
Rotation dynamics
At the beginning of the article, we recorded the kinematic characteristics that are used to describe the motion of rotation. In the dynamics of rotation, the main equation that uses these characteristics is the following:
M = I * α.
The action of the moment M on the system having the moment of inertia I leads to the appearance of angular acceleration α.
This formula is used to determine the angular frequencies of rotation in the technique. For example, knowing the torque of an induction motor, which depends on the frequency of the current in the stator coil and the magnitude of the changing magnetic field, as well as knowing the inertial properties of the rotating rotor, it is possible to determine to what speed ω the motor rotor spins in a known time t.
Problem solving example
The weightless lever, whose length is 2 meters, has a support in the middle. What weight should be put on one end of the lever so that it is in a state of equilibrium, if on the other side of the support at a distance of 0.5 meters from it lies a load weighing 10 kg?
Obviously, the balance of the lever will come if the moments of forces created by the loads are equal in absolute value. The force that creates the moment in this task is the weight of the body. The leverage is equal to the distance from the load to the support. We write the corresponding equality:
M 1 = M 2 =>
m 1 * g * d 1 = m 2 * g * d 2 =>
P 2 = m 2 * g = m 1 * g * d 1 / d 2 .
We obtain the weight of P 2 if we substitute the values m 1 = 10 kg, d 1 = 0.5 m, d 2 = 1 m from the conditions of the problem. The written equality gives the answer: P 2 = 49.05 Newton.