Rotational dynamics is one of the important branches of physics. It describes the causes of the movement of bodies in a circle around a certain axis. One of the important values of the dynamics of rotation is the moment of force, or torque. What is a moment of power? Consider this concept in this article.
What you should know about the rotation of bodies?
Before answering the question that this is a moment of force, we characterize the rotation process from the point of view of physical geometry.
Each person intuitively understands what is at stake. Rotation implies such a motion of a body in space, when all its points move along circular paths around a certain axis or point.
Unlike linear displacement, the rotation process is described by angular physical characteristics. Among them, the rotation angle θ, the angular velocity ω, and the angular acceleration α should be mentioned. The value of θ is measured in radians (rad), ω in rad / s, and α in rad / s 2 .
Examples of rotation are the movement of our planet around its star, the spinning of an engine rotor, the movement of a ferris wheel, and others.
The concept of torque
The moment of force is a physical quantity equal to the vector product of the radius vector r¯ directed from the axis of rotation to the point of application of the force F¯ and the vector of this force. Mathematically, this is written as follows:
M¯ = [r¯ * F¯].
As you can see, the moment of force is a vector quantity. Its direction is determined by the rule of a gimlet or right hand. The quantity M¯ is directed perpendicular to the plane of rotation.
In practice, it often becomes necessary to calculate the absolute value of the moment M¯. To do this, use the expression:
M = r * F * sin (φ).
Where φ is the angle between the vectors r¯ and F¯. The product of the modulus of the radius vector r and the sine of the marked angle is called the arm of force d. The latter is the distance between the vector F¯ and the axis of rotation. The formula above can be rewritten as:
M = d * F, where d = r * sin (φ).
The moment of force is measured in Newtons per meter (N * m). Nevertheless, one should not resort to the use of joules (1 N * m = 1 J), since the quantity M¯ is not a scalar, but a vector.
The physical meaning of M¯
The physical meaning of the moment of force is easiest to understand with the following examples:
- We propose to do the following experiment: try to open the door by pushing it near the hinges. To do this operation with success, you will have to make great efforts. At the same time, any door opens easily by the handle. The difference between the two cases described is the length of the shoulder of the force (in the first case it is very small, therefore, the created moment will be small and requires the application of great force).
- Another experiment showing the meaning of the torque is this: take a chair and try to keep it on your outstretched arm on the weight. This is difficult to do. At the same time, if you press your hand with a chair to the body, then the task will no longer seem overwhelming.
- Everyone associated with technology knows that it is much easier to unscrew a nut with a wrench than with your fingers.
All these examples speak of one thing: the moment of force reflects the ability of the latter to rotate the system around its axis. The greater the torque, the higher the likelihood that he will perform a turn in the system and give it angular acceleration.
Torque and body balance
Statics - a section that studies the causes of the equilibrium of bodies. If the system under consideration has one or several axes of rotation, then this system can potentially make a circular motion. To prevent this from happening and the system was at rest, the sum of all n external moments of forces relative to any axis should be zero, that is:
∑ i = 1 n M i = 0.
When using this condition for equilibrium of bodies during the solution of practical problems, it should be remembered that any force striving to rotate the system counterclockwise creates a positive torque, and vice versa.
Obviously, if a force is applied to the axis of rotation, then it will not create any moment (shoulder d is zero). Therefore , the reaction force of the support never creates a moment of force, if it is calculated relative to this support.
Task example
Having figured out how to determine the moment of force, we will solve the following interesting physical problem: suppose that there is a table on two supports. The table is 1.5 meters long and weighs 30 kg. A load of 5 kg was placed at a distance of 1/3 from the right edge of the table. It is necessary to calculate what reaction force will act on each table support with a load.
The calculation of the problem should be carried out in two stages. At first, consider a table without a load. Three forces act on it: two identical reactions of support and body weight. Since the table is symmetrical, the reactions of the supports are equal to each other and together balance the weight. The value of each support reaction is:
N 0 = P / 2 = m * g / 2 = 30 * 9.81 / 2 = 147.15 N.
As soon as they put the load on the table, the reaction values of the supports change. To calculate them, we use the balance of moments. First, we consider the moments of forces acting relative to the left table support. There are two of these points: the additional reaction of the right support without taking into account the weight of the table and the weight of the load itself. Since the system is in equilibrium, we obtain:
ΔN 1 * l - m 1 * g * 2/3 * l = 0.
Here l is the length of the table, m 1 is the mass of the cargo. From the expression we get:
ΔN 1 = m 1 * g * 2/3 = 2/3 * 9.81 * 5 = 32.7 N.
In a similar way, we calculate the additional reaction to the left table support. We get:
-ΔN 2 * l + m 1 * g * 1/3 * l = 0;
ΔN 2 = m 1 * g * 1/3 = 1/3 * 5 * 9.81 = 16.35 N.
To calculate the reaction of table supports with a load, it is necessary to add ΔN 1 and ΔN 2 to N 0 , we obtain:
right support: N 1 = N 0 + ΔN 1 = 147.15 + 32.7 = 179.85 N;
left support: N 2 = N 0 + ΔN 2 = 147.15 + 16.35 = 163.50 N.
Thus, the load on the right table support will be more than on the left.