If the linear movement of bodies is described in classical mechanics using Newton’s laws, then the characteristics of the motion of mechanical systems along circular paths are calculated using a special expression called the equation of moments. What points are we talking about and what is the meaning of this equation? These and other issues are disclosed in the article.
Moment of power
Everyone is well aware of the Newtonian force, which, acting on the body, leads to the message of acceleration. When such a force is applied to an object that is fixed on a certain axis of rotation, then this characteristic is usually called the moment of force. The equation of the moment of force can be written in the following form:
M¯ = L¯ * F¯
A graphic explaining this expression is shown below.
It can be seen here that the force F¯ is directed towards the vector L¯ at an angle Φ. The vector L¯ itself is assumed to be directed from the axis of rotation (indicated by an arrow) to the point of application F¯.
The above formula is a product of two vectors; therefore, the quantity M¯ is also directional. Where will the moment of force M¯ be turned? This can be determined by the rule of the right hand (four fingers are directed along the trajectory from the end of the vector L¯ to the end of F¯, and the set apart thumb shows the direction of M¯).
In the figure above, the expression for the moment of force in scalar form will take the form:
M = L * F * sin (Φ)
If you carefully look at the figure, you can see that L * sin (Φ) = d, then we have the formula:
M = d * F
The value of d is an important characteristic in calculating the moment of force, since it reflects the efficiency of the applied F to the system. This value is usually called the lever of force.
The physical meaning of M lies in the ability of a force to rotate the system. Everyone can feel this ability if they open the door by the handle, pushing it near the hinges, or try to unscrew the nut with a short and long key.
System balance
The concept of a moment of force is very useful when considering the equilibrium of a system that is subject to several forces and that has an axis or point of rotation. In such cases, apply the formula:
∑ i M i ¯ = 0
That is, the system will be in equilibrium if the sum of all the moments of forces applied to it is zero. Note that in this formula there is a sign of the vector above the moment, that is, when deciding, one should not forget to take into account the sign of this quantity. It is generally accepted that the acting force that rotates the system counterclockwise creates a positive M i ¯.
A striking example of the tasks of this type are problems with the balance of leverage of Archimedes.
Momentum
This is another important characteristic of circular motion. In physics, it is described as the product of the amount of movement by the lever. The momentum equation has the following form:
T¯ = r¯ * p¯
Here p¯ is the momentum vector, r¯ is the vector connecting the rotating material point with the axis.
The figure explaining this expression is shown below.
Here ω is the angular velocity, which will further appear in the equation of moments. Note that the direction of the vector T¯ is found by the same rule as M¯. In the figure above, T¯ in direction will coincide with the angular velocity vector ω¯.
The physical meaning of T¯ is the same as the p¯ characteristics in the case of linear motion, that is, the angular momentum describes the amount of rotational motion (stored kinetic energy).
Moment of inertia
The third important characteristic, without which it is impossible to compose the equation of motion of a rotating object, is the moment of inertia. It appears in physics as a result of mathematical transformations of the formula for the angular momentum of a material point. We show how this is done.
We represent the value of T¯ in the following form:
T¯ = r¯ * m * v¯, where p¯ = m * v¯
Using the relationship between angular and linear velocities, we can rewrite this expression as follows:
T¯ = r¯ * m * r¯ * ω¯, where v¯ = r¯ * ω¯
We write the last expression in the form:
T¯ = r 2 * m * ω¯
The value of r 2 * m is the moment of inertia I for a point of mass m, which makes a circular motion around the axis at a distance from it r. This particular case allows us to introduce the general equation of moment of inertia for a body of arbitrary shape:
I = ∫ m (r 2 * dm)
I is an additive quantity, the meaning of which is the inertia of a rotating system. The larger I, the more difficult it is to unwind the body, and significant efforts must be made to stop it.
Equation of moments
We examined three quantities whose name begins with the word "moment". This was done on purpose, since they are all connected in one expression, called the equation of 3 moments. Derive it.
Consider the expression for the angular momentum T¯:
T¯ = I * ω¯
Find how the value of T¯ changes over time, we have:
dT¯ / dt = I * dω¯ / dt
Given that the derivative of the angular velocity is equal to that for the linear velocity divided by r, and also revealing the value of I, we arrive at the expression:
dT¯ / dt = m * r 2 * 1 / r * dv¯ / dt = r * m * a¯, where a¯ = dv¯ / dt is linear acceleration.
Note that the product of mass and acceleration is nothing more than the acting external force F¯. As a result, we get:
dT¯ / dt = r * F¯ = M¯
We came to an interesting conclusion: the change in angular momentum is equal to the moment of the acting external force. This expression is usually written in a slightly different form:
M¯ = I * α¯, where α¯ = dω¯ / dt is the angular acceleration.
This equality is called the equation of moments. It allows you to calculate any characteristic of a rotating body, knowing the parameters of the system and the magnitude of the external impact on it.
Conservation Law T¯
The conclusion obtained in the previous paragraph indicates that if the external moment of forces is equal to zero, then the angular momentum will not change. In this case, we write the expression:
T¯ = const. or I 1 * ω 1 ¯ = I 2 * ω 2 ¯
This formula is called the law of conservation of T¯. That is, any changes within the system do not change the total angular momentum.
This fact is used by skaters and ballerinas during their performances. It is also used if it is necessary to perform a rotation around its axis of an artificial satellite moving in space.