In the general course of physics, the two most simple types of moving objects in space are studied - this is translational motion and rotation. If the dynamics of translational motion is based on the use of such quantities as forces and masses, then the concepts of moments are used to quantify the rotation of bodies. In this article, we will consider by what formula the moment of force is calculated, and for solving what problems this quantity is used.
Moment of power
Imagine a simple system that consists of a material point rotating around an axis at a distance r from it. If tangential force F is applied to this point, which is perpendicular to the axis of rotation, then it will lead to the appearance of angular acceleration of the point. The ability of a force to rotate a system is called a torque or torque. Calculated by the formula of the following:
M¯ = [r¯ * F¯]
In square brackets is the vector product of the radius vector and the force. The radius vector r¯ is a directed segment from the axis of rotation to the point of application of the vector F¯. Given the property of a vector product, for the value of the moment modulus, the formula in physics is written in the following form:
M = r * F * sin (φ) = F * d, where d = r * sin (φ).
Here, the angle between the vectors r¯ and F¯ is denoted by the Greek letter φ. The value of d is called the shoulder of strength. The larger it is, the more torque can be created by force. For example, if you open the door by pressing on it near the hinges, then the shoulder d will be small, so you need to apply great force to rotate the door on the hinges.
As can be seen from the moment formula, the quantity M¯ is a vector. It is directed perpendicular to the plane in which the vectors r¯ and F¯ lie. The direction of M¯ is easy to determine using the rule of the right hand. To use it, you need to direct four fingers of the right hand along the vector r¯ in the direction of the force F¯. Then the bent thumb will show the direction of the moment of force.
Static moment of power
The considered value is very important in calculating the equilibrium conditions for a system of bodies having a rotation axis. There are only two such conditions in statics:
- equality to zero of all external forces that have one or another effect on the system;
- equality to zero of the moments of forces associated with external forces.
Both equilibrium conditions can be mathematically written as follows:
∑ i (F i ¯) = 0;
∑ i (M i ¯) = 0.
As can be seen, it is necessary to calculate exactly the vector sum of the quantities. As for the moment of force, it is considered to be its positive direction if the force makes a turn against the clock hands. Otherwise, a minus sign should be used before the moment determination formula.
Note that if the rotation axis in the system is located on some support, then the corresponding reaction force of the moment does not create, since its shoulder is equal to zero.
The moment of power in dynamics
The dynamics of rotation around the axis has, like the dynamics of translational displacement, the basic equation on the basis of which many practical problems are solved. It is called the equation of moments. The corresponding formula is written as:
M = I * α.
In fact, this expression is Newton’s second law, if the moment of force is replaced by force, the moment of inertia I is mass, and the angular acceleration α is replaced by a similar linear characteristic. To better understand this equation, we note that the moment of inertia plays the same role as the ordinary mass in translational motion. The moment of inertia depends on the distribution of mass in the system relative to the axis of rotation. The greater the distance of the body to the axis, the greater the value of I.
The angular acceleration α is calculated in radians per second squared. It characterizes the speed of change of rotation.
If the moment of force is zero, then the system does not receive any acceleration, which indicates the conservation of its angular momentum.
Work of the moment of force
Since the studied value is measured in Newtons per meter (N * m), many may think that it can be replaced by a joule (J). However, they do not do this because a certain energy quantity is measured in joules, while the moment of force is a force characteristic.
As well as force, moment M can also do the work. It is calculated by the following formula:
A = M * θ.
Where the Greek letter θ denotes the angle of rotation in radians, which the system rotated as a result of the action of the moment M. Note that as a result of multiplying the moment of force by the angle θ, the units are saved, however, the units of work, that is, Joules, are already used.