Movement around the axis of rotation is one of the common types of movement of objects in nature. In this article, we consider this type of motion from the point of view of dynamics and kinematics. We also give formulas that relate the basic physical quantities.
What kind of movement will we talk about?
In the literal sense, we will talk about moving bodies around a circle, that is, about their rotation. A striking example of such a movement is the rotation of the wheel of a car or bicycle while moving a vehicle. A rotation around its axis of a skater performing complex pirouettes on ice. Or the rotation of our planet around the Sun and around its own axis inclined to the plane of the ecliptic.
As you can see, an important element of this type of movement is the axis of rotation. Each point of a body of arbitrary shape makes circular movements around it. The distance from a point to an axis is called the radius of rotation. Many properties of the entire mechanical system depend on its value, for example, the moment of inertia, linear velocity, and others.
Rotation dynamics
If the cause of the linear translational movement of bodies in space is the external force acting on them, then the cause of motion around the axis of rotation is the external moment of force. This quantity is described as the vector product of the applied force F¯ by the distance vector from the point of its application to the r¯ axis, that is:
M¯ = [r¯ * F¯]
The action of the moment M¯ leads to the appearance of angular acceleration α¯ in the system. Both quantities are related to each other through a certain coefficient I by the following equality:
M¯ = I * α¯
The value of I is called the moment of inertia. It depends both on the shape of the body and on the distribution of mass inside it and on the distance to the axis of rotation. For a material point, it is calculated by the formula:
I = m * r 2
If the external angular momentum is zero, then the system retains its angular momentum L¯. This is another vector quantity, which, by definition, is equal to:
L¯ = [r¯ * p¯]
Here p¯ is a linear impulse.
The law of conservation of moment L¯ is usually written as follows:
I * ω = const
Where ω is the angular velocity. We will discuss this later in the article.
Kinematics of rotation
Unlike dynamics, this branch of physics considers exclusively practical important quantities associated with a change in time of the position of bodies in space. That is, the object of studying the kinematics of rotation are speeds, accelerations and rotation angles.
First, we introduce the angular velocity. By it is meant the angle at which the body makes a turn in a unit of time. The formula for the instantaneous angular velocity is:
ω = dθ / dt
If for equal periods of time the body makes turns at equal angles, then the rotation is called uniform. The formula for the average angular velocity is valid for him:
ω = Δθ / Δt
Ω is measured in radians per second, which in the SI system corresponds to the inverse of seconds (s -1 ).
In the case of uneven rotation, the concept of angular acceleration α is used. It determines the rate of change in time of the quantity ω, that is:
α = dω / dt = d 2 θ / dt 2
Α is measured in radians per second square (in SI - s -2 ).
If the body initially rotated uniformly with a speed of ω 0 , and then began to increase its speed with a constant acceleration α, then such a movement can be described by the following formula:
θ = ω 0 * t + α * t 2/2
This equality is obtained by integrating the angular velocity equations over time. The formula for θ allows you to calculate the number of revolutions that the system will make around the axis of rotation in time t.
Linear and angular speeds
Both speeds are connected to each other. When talking about the speed of rotation around the axis, they can mean both linear and angular characteristics.
Suppose that some material point rotates around an axis at a distance r with a speed ω. Then its linear velocity v will be equal to:
v = ω * r
The difference between linear and angular velocity is significant. So, ω with uniform rotation does not depend on the distance to the axis, but the quantity v linearly increases with increasing r. The latter fact explains why, with an increase in the radius of rotation, it is more difficult to keep the body on a circular path (its linear velocity and, as a result, inertial forces increase).
The task of calculating the speed of rotation around its axis of the Earth
Everyone knows that our planet in the solar system performs two types of rotational motion:
- around its axis;
- around the star.
We calculate the velocities ω and v for the first of them.
Angular velocity is not difficult to determine. To do this, we recall that the planet completes a complete revolution equal to 2 * pi radians in 24 hours (the exact value is 23 hours 56 minutes, 4.1 seconds). Then the value of ω will be equal to:
ω = 2 * pi / (24 * 3600) = 7.27 * 10 -5 rad / s
The calculated value is small. Let us show how strongly the absolute value of ω differs from that for v.
We calculate the linear velocity v for points lying on the surface of the planet at the latitude of the equator. Since the Earth is a flattened ball, the equatorial radius is slightly larger than the polar one. It is 6378 km. Using the formula for the connection of two speeds, we obtain:
v = ω * r = 7.27 * 10 -5 * 6378000 ≈ 464 m / s
The resulting speed is 1670 km / h, which is greater than the speed of sound in air (1235 km / h).
The rotation of the Earth around its axis leads to the appearance of the so-called Coriolis force, which should be taken into account when flying ballistic missiles. It is also the cause of many atmospheric phenomena, for example, deviations of the direction of the winds of the trade winds to the west.