The movement of bodies around the circumference is quite common in our life and in nature. Vivid examples of this type of movement are the rotation of windmills, planets around their stars and wheels of vehicles. In this article, we consider what formulas describe the motion around the circle of bodies.
Movement in a circle and in a straight line in physics
In physics, kinematics deals with motion issues. It establishes a connection between the quantities describing this process. In dynamics, attention is also paid to movement, but it is focused on a description of the causes of its occurrence. In other words, if for kinematics the main physical quantities are the path and speed, then for dynamics these are the forces acting on the bodies.
In physics, it is customary to distinguish two ideal types of motion trajectories:
The mathematical apparatus for describing motion along both types of trajectories is so well developed that an understanding of formulas, for example for rectilinear motion, automatically leads to an understanding of the expressions for circular motion. The only fundamental difference between the formulas of these types of displacement is that it is convenient to use angular characteristics rather than linear ones to move around the circle.
Further in the article we will consider exclusively kinematic formulas of motion around the circumference of bodies, without going into details of the dynamics.
Angular motion characteristics: rotation angle
Before writing down the formulas for circular motion in physics, one should introduce the quantities that will appear in these formulas.
Let's start with the rotation angle. We will denote it by the Greek letter θ (theta). Since rotation involves the movement of a point along the same circle, the value of the rotation angle θ for a certain period of time can be used to determine the number of revolutions that this point made. Recall that the entire circle is 2 * pi radians, or 360 o . Then the formula for the number of revolutions n through the angle θ takes the form:
n = θ / (2 * pi)
Hereinafter, in all formulas, the angle is expressed in radians.
Using the known angle θ, it is also possible to determine the linear distance that a point has traveled along a circle. This distance will be equal to:
L = θ * r
Here r is the radius of the circle under consideration.
Angular velocity and acceleration
Kinematic formulas for moving around a point's circle also involve the use of the concepts of angular velocity and angular acceleration. We denote the first letter by ω (omega) and the second by α (alpha).
The physical meaning of the angular velocity ω is simple: this value shows what angle in radians a point rotates for every second of time. This definition has the following mathematical representation:
ω = dθ / dt
This circle velocity formula is written in differential form. The value of ω obtained with its help is called the instantaneous velocity. It is convenient to use if the movement is not uniform, that is, it occurs at a variable speed.
The angular acceleration α is a quantity that describes the rate of change of speed ω, that is:
α = dω / dt
Angular acceleration is measured in radians per second square (rad / s 2 ). So, 1 rad / s 2 means that the body increases the speed by 1 rad / s for every second of time.
Given the expression for ω written above, the equality can be represented in the following form:
α = d 2 θ / dt 2
Depending on the features of circular motion, the value of α can be constant, variable, or zero.
Uniform movement
When no external force acts on the rotating body, the angular velocity will remain constant for an arbitrarily long time. This movement is called uniform rotation. It is described by the following formula:
θ = ω * t
In this expression, only two quantities are variables: t and θ. Speed ω = const.
One important point should be noted: only the resultant of external forces on the body is equal to zero, while the internal forces acting in the system are not equal to zero. So, the internal force causes the rotating body to change its rectilinear path to a curved (circle). This force gives rise to centripetal acceleration. The latter does not change either the velocity ω or the linear velocity v, it only changes the direction of motion.
Equally accelerated circumferential movement
The formulas for this type of displacement can be obtained directly from the above mathematical expressions for the quantities ω and α. Equally accelerated motion assumes that over a more or less long period of time, the modulus and direction of acceleration α do not change. Due to this, we can integrate the differential expression for α and obtain the following two formulas:
ω = ω 0 + α * t;
ω = ω 0 - α * t
Obviously, in the first case, the movement will be equally accelerated, in the second - equally slow. The value of ω 0 here is a certain initial velocity that a rotating body possessed before acceleration appeared.
For uniformly accelerated motion, there is no finite velocity, since it can increase arbitrarily long. For equally slow motion, the final state will be the cessation of rotation, i.e., ω = 0.
Now we write the formulas for determining the angle θ when moving with constant acceleration. These formulas are obtained by double integration over time for the expression of α in terms of θ. The following expressions are obtained:
θ = ω 0 * t + α * t 2/2;
θ = ω 0 * t - α * t 2/2
That is, the central angle θ, which the body rotates in time t, will be equal to the sum of the two terms. The first term is the contribution to θ of uniform motion, the second is uniformly accelerated (equally slow).
The relationship between angular and linear quantities
In considering the concept of the angle of rotation θ, a formula has already been given that relates it to the linear distance L. Here we also consider similar expressions for the velocity ω and acceleration α.
The linear velocity v with uniform motion is defined as the distance L traveled during time t, that is:
v = L / t
Substituting here the expression for L through θ, we obtain:
v = L / t = θ / t * r = ω * r
We got the relationship between linear and angular velocity. It is important to note that the convenience of using angular velocity is due to the fact that it does not depend on the radius of the circle. In turn, the linear velocity v increases linearly with increasing r.
It remains to write the relationship between linear acceleration a and its angular analogue α. To do this, we write the expression for the velocity v with uniformly accelerated motion without the initial velocity v 0 . We get:
v = a * t
We substitute here the obtained expression of the relation between v and ω:
ω * r = a * t =>
a = ω / t * r = α * r
Like speed, linear acceleration tangential to a circle depends on the radius.
Centripetal acceleration
We have already said a few words about this quantity. Here are the formulas that can be used to calculate it. Through velocity v, the expression for centripetal acceleration a c has the form:
a c = v 2 / r
Through angular velocity it can be written as follows:
a c = ω 2 * r 2 / r = ω 2 * r
The value of a c has nothing to do with tangential acceleration a. Centripetal acceleration maintains a rotating body on one circle.
The task of determining the angular velocity of rotation of the planet
It is known that Mercury is closest to the sun. Assuming that it rotates around a circle around the body, we can determine its angular velocity ω.
To solve the problem, refer to the reference data. From them it is known that the planet makes a complete revolution around the luminary in 87 days 23.23 earthly hours. This time is called the circulation period. Given that the movement occurs at a constant angular velocity, we write the working formula:
θ = ω * t =>
ω = θ / t
It remains to translate the time into seconds, substitute the value of the angle θ corresponding to the total revolution (2 * pi), and write down the answer: ω = 8.26 * 10 -7 rad / s.