In nature and technology, we often encounter the manifestation of the rotational motion of solids, for example, shafts and gears. As this type of motion is described in physics, what formulas and equations are used for this, these and other questions are covered in this article.
What is rotation?
Each of us intuitively knows what kind of movement will be discussed. Rotation is a process in which a body or a material point moves in a circular path around a certain axis. From a geometric point of view, the axis of rotation of a solid is a straight line, the distance to which remains unchanged during the movement. This distance is called the radius of rotation. We will denote it by the letter r. If the axis of rotation passes through the center of mass of the body, then it is called its own axis. An example of rotation around its own axis is the corresponding motion of the planets of the solar system.
For rotation to occur, there must be centripetal acceleration, which arises due to centripetal force. This force is directed from the center of mass of the body to the axis of rotation. The nature of the centripetal force can be very different. So, on a cosmic scale, gravity plays its role, if the body is fixed by a thread, then the tension force of the latter will be centripetal. When the body rotates around its own axis, the role of the centripetal force is played by the internal electrochemical interaction between the elements making up the body (molecules, atoms).
It must be understood that without the presence of centripetal force, the body will move in a straight line.
Physical quantities describing rotation
Firstly, these are dynamic characteristics. These include:
- angular momentum L;
- moment of inertia I;
- moment of force M.
Secondly, these are kinematic characteristics. We list them:
- rotation angle θ;
- angular velocity ω;
- angular acceleration α.
We briefly describe each of these quantities.
The moment of momentum is determined by the formula:
L = p * r = m * v * r
Where p is the linear momentum, m is the mass of the material point, v is its linear velocity.
The moment of inertia of the material point is calculated using the expression:
I = m * r 2
For any body of complex shape, the value of I is calculated as the integral sum of the moments of inertia of the material points.
The moment of force M is calculated as follows:
M = F * d
Here F is an external force, d is the distance from the point of its application to the axis of rotation.
The physical meaning of all quantities in the name of which the word "moment" is present is similar to the meaning of the corresponding linear quantities. For example, the moment of force indicates the ability of the applied force to report angular acceleration to a system of rotating bodies.
Kinematic characteristics are mathematically determined by the following formulas:
ω = dθ / dt;
α = dω / dt.
As can be seen from these expressions, the angular characteristics in their meaning are linear (speed v and acceleration a), only they are applicable for a circular path.
Rotation dynamics
In physics, the study of the rotational motion of a solid is carried out using two branches of mechanics: dynamics and kinematics. Let's start with the dynamics.
Dynamics studies external forces acting on a system of rotating bodies. Immediately write the equation of rotational motion of a solid, and then, we analyze its components. So, this equation has the form:
M = I * α
The moment of force that acts on the system with the moment of inertia I causes the appearance of angular acceleration α. The smaller the value of I, the easier it is to, with the help of a certain moment M, spin up the system to high speeds in small time intervals. For example, a metal rod is easier to rotate along its axis than perpendicular to it. However, the same rod is easier to rotate around an axis perpendicular to it and passing through the center of mass than through its end.
The law of conservation of L
Above this value was introduced, it is called the angular momentum. The equation of the rotational motion of a solid body, presented in the previous paragraph, is often written in a different form:
M * dt = dL
If the moment of external forces M acts on the system during the time dt, then it causes a change in the angular momentum of the system by the value dL. Accordingly, if the moment of forces is zero, then L = const. This is the law of conservation of L. For her, using the relationship between linear and angular velocity, we can write:
L = m * v * r = m * ω * r 2 = I * ω.
Thus, in the absence of a moment of forces, the product of the angular velocity and the moment of inertia is a constant. This physical law is used by skaters in their performances or artificial satellites, which must be turned around their own axis in outer space.
Centripetal Acceleration
Above, when studying the rotational motion of a solid, this quantity has already been described. The nature of the centripetal forces was also noted. Here we just supplement this information and give the corresponding formulas for calculating this acceleration. Denote it a c .
Since the centripetal force is directed perpendicular to the axis and passes through it, it does not create a moment. That is, this force has absolutely no effect on the kinematic characteristics of rotation. However, it creates centripetal acceleration. We give two formulas for its definition:
a c = v 2 / r;
a c = ω 2 * r.
Thus, the greater the angular velocity and radius, the greater the force should be applied in order to keep the body on a circular path. A striking example of this physical process is the skidding of a car during a turn. Skidding occurs if the centripetal force, whose role is played by the friction force, becomes less than the centrifugal force (inertial characteristic).
Kinematics of rotation
Three basic kinematic characteristics were listed above in the article. The kinematics of the rotational motion of a solid body are described by the formulas as follows:
θ = ω * t => ω = const., α = 0;
θ = ω 0 * t + α * t 2/2 => ω = ω 0 + α * t, α = const.
The first line contains formulas for uniform rotation, which implies the absence of an external moment of forces acting on the system. The second line contains formulas for uniformly accelerated circular motion.
Note that rotation can occur not only with positive acceleration, but also with negative. In this case, in the formulas of the second line, put a minus sign in front of the second term.
Problem solving example
A force moment of 1000 N * m acted on a metal shaft for 10 seconds. Knowing that the moment of inertia of the shaft is 50 kg * m 2 , it is necessary to determine the angular velocity that the mentioned moment of force gave to the shaft.
Using the basic equation of rotation, we calculate the acceleration of the shaft:
M = I * α =>
α = M / I.
Since this angular acceleration acted on the shaft for a time t = 10 seconds, we use the formula for uniformly accelerated motion to calculate the angular velocity:
ω = ω 0 + α * t = M / I * t.
Here ω 0 = 0 (the shaft did not rotate before the action of the moment of forces M).
We substitute the numerical values of the quantities into the equality, we obtain:
ω = 1000/50 * 10 = 200 rad / s.
In order to translate this number into familiar revolutions per second, it is necessary to divide it by 2 * pi. Performing this action, we obtain that the shaft will rotate at a frequency of 31.8 rpm./s.