Each student knows what a circle is and what parameters it is characterized by. Circular motion is one of the most common types of movement of objects in nature. That this is a uniform movement around the circle, and what formulas it describes, we will tell in this article.
Rotation motion
To more clearly understand what is at stake, we will immediately give examples of this type of movement of bodies in space. Examples are:
- metal shaft rotation;
- movement of the wheel of a car or bicycle;
- unwinding of a stone tied to a rope;
- rotation of the toy-yula;
- the movement of our planet around the sun and around its own axis.
The elements that characterize the type of movement in question are:
- A circle that has a certain radius r. The circle is the trajectory of movement.
- The axis around which the rotation occurs. It is located in the center of the circle in a plane perpendicular to it.
In physics, a set of kinematic quantities is used to describe any movement of objects. Next, we present these values for rotational motion.
Angular rotation characteristics
These include the angle of rotation θ, angular velocity ω, and angular acceleration α. In physics of uniform circular motion, these three quantities take on a certain value.
As for the angle θ, its physical meaning is intuitive. The value of θ shows the central angle of the circular sector, by which the body rotates in a known period of time. Further in the formulas it is expressed in radians. Recall that the full circle corresponds to 2 * pi radians.
The speed ω is an analog of a linear quantity, only its units are radians per second. In differential form, the mathematical definition of ω looks like this:
ω = dθ / dt
Angular velocity shows the speed of rotation of the body. It can be used to calculate the speed f in revolutions per second. To do this, use the formula:
f = ω / (2 * pi)
Acceleration α shows how quickly the rotation speed ω changes. That is, the expression is true for him:
α = dω / dt
The unit of measure for α is the radians per square second (rad / s 2 ).
What is a uniform circular motion?
In physics, grades 10 study the topic in detail. Everyone knows that the rectilinear motion will be uniform if the linear velocity v of the body remains constant. In the case of rotational motion, the following definition can be given: uniform circular motion is the rotation of a body with a constant angular velocity. Since the latter does not change, the angular acceleration α is equal to zero. The above reasoning allows us to write the basic formula that describes the type of movement of bodies under consideration:
θ = ω * t
In the expression, ω = const, and the quantities θ and t change. A graphic representation of this type of movement is a straight line that passes through the origin. The tangent of the angle between this line and the x axis is equal to the absolute value of the angular velocity ω.
A feature of uniform circumferential movement is the passage by the body over equal intervals of time of equal angular segments. This feature distinguishes uniform rotation from accelerated.
Linear and angular velocity with uniform circular motion
Until now, the kinematics of rotational motion has been considered within the framework of angular characteristics. Their use is convenient for solving practical problems. Nevertheless, sometimes it is useful to know the linear characteristics of a rotating system. An important of these is the linear velocity v.
To find the relationship between the quantities v and ω, we assume that the rotation of the material point is carried out around a circle of radius r. Then the linear velocity with uniform movement around the circle is a value that is calculated by the formula:
v = 2 * pi * r / t
Where t is the time the body makes one revolution around the axis of rotation. The angular velocity ω can be calculated as follows:
ω = 2 * pi / t
Comparing these expressions, we arrive at the formula:
v = ω * r
Thus, the linear velocity for a material point rotating in a circular trajectory with radius r is a constant value. However, if the rotation of a massive body is considered, then the linear velocity of each of its points will increase with increasing distance to the axis of rotation, while the value of ω remains constant.
Centripetal Acceleration
For the body to be able to rotate around the circumference, the presence of some force is necessary, which is internal with respect to the rotating system. This force provides a curvature of the rectilinear trajectory of motion of any body with inertial properties. It is a centripetal force, which is directed from the center of mass of the body to the axis of rotation. Uniform circular motion and centripetal motion are one and the same process. If there were no centripetal force, then rotation would be impossible.
According to Newton's laws, the action on the body of any force leads to the appearance of acceleration. With uniform motion around the circle, there is no angular acceleration (it is equal to zero), however, there is centripetal acceleration. It is calculated by the formula:
a c = v 2 / r
Acceleration a c does not lead to a change in the modulus of velocities v and ω. The result of its action is a change in the direction of the vector v.
The solution of the problem
Assuming that the motion of our planet around the Sun is uniform, it is necessary to find the value of the centripetal acceleration with which solar gravity acts on the Earth.
Using the angular velocity, we can write the following formula for a c :
a c = ω 2 * r
In turn, the value of ω for the Earth is determined by the formula:
ω = 2 * pi / T
Then we obtain for centripetal acceleration a c :
a c = 4 * pi 2 * r / T 2
Where r is the average distance from the earth to the sun, T is the earthly year. Given that r = 149.6 million km, T = 365.3 days, we get the answer: a c = 0.006 m / s 2 . This value is hundreds of times smaller than typical linear accelerations when driving vehicles.