The distance and time it takes to overcome this distance, connects the physical concept of speed. Humans, as a rule, not questioned the determination of this value. Everyone understands that to move the vehicle at a speed of 100 km/h means one hour to travel 100 km.
But what if the body is rotating? For example, a typical domestic fan makes ten revolutions per second. And at the same time, the rotation speed of the blades is such that they can be easily stopped by hand without harm to themselves. The earth around its star – the Sun makes one revolution in a year, and that more than 30 million seconds, but its velocity in the circumstellar orbit is about 30 kilometers per second!
How to tie the usual speed with the speed of rotation, is the formula of angular velocity?
The concept of angular velocity
The concept of angular velocity used in the study of the laws of rotation. It applies to all rotating bodies. Whether the rotation of a certain mass around the other, as in the case of the earth and the Sun, or the rotation of the body around the polar axis (diurnal rotation of our planet).
The difference of the angular velocity from the linear in that it captures the change in the angle, not distance per unit time. In physics, angular velocity is denoted by the Greek letter omega – ω.
The classical formula for angular speed is considered.
Imagine that around a certain center And a physical body rotates with a constant speed. Its position in space relative to the center is determined by the angle φ. At some time t1 the considered body is in the point B. the Angle of deviation of the body from the initial φ1.
Then the body moves to point C. It is there at time t2. The time it took for a given move:
The position of the body in space also changes. Now the deflection angle is φ2. The change in angle over time Δt was:
∆φ = φ2 – φ1.
Now the formula for angular velocity is formulated as follows: the angular velocity is defined as the attitude angle change ∆φ during ∆t.
Units of measurement of angular velocity
The speed of the linear body is measured in different units. The movement of vehicles on the roads habitually indicate in miles per hour, ships make knots – nautical miles per hour. If we consider the movement of space bodies, there often appear a kilometers per second.
The angular velocity depending on the magnitude and on the subject that is rotating is also measured in different units.
Radians per second (rad/s) is a classical measure of speed in the international system of units (SI). Show – for how many radians (one complete revolution of 2 ∙ 3.14 radians) manages to turn the body in one second.
Revolutions per minute (rpm) - the most common unit for indicating the speed of rotation in engineering. The shafts of the engines, both electric and automotive, are given out (just look at the tachometer in your car), rpm.
Revolutions per second (R/s) – used rarely, primarily for educational purposes.
Period of circulation
Sometimes to determine the speed of rotation is more convenient to use another concept. The orbital period is called a time for which a body rotates 360° (full circle) around the center of rotation. The formula of angular speed, expressed in terms of orbital period, takes the form:
To Express a period of the speed of rotation of the bodies is justified in cases when a body rotates relatively slowly. Let us return to the consideration of the motion of our planet around the sun.
The angular velocity formula allows us to calculate it, knowing the period of revolution:
Looking at the result, one can understand why, considering the rotation of celestial bodies, it is more convenient to use the period of circulation. A person sees before him comprehensible figures and clearly visualizes their scale.
The relationship between angular and linear velocities
Some tasks must be defined linear and angular velocity. The formula for the transformation is simple: the linear velocity of a body is equal to the product of the angular velocity on the radius of rotation. As shown in the figure.
The expression also "works" in the reverse order, with its help the angular velocity is also determined. The formula through speed linear turns out by simple arithmetic manipulations.