Usually, when we talk about movement, we imagine an object that moves in a straight line. The speed of such movement is usually called linear, and the calculation of its average value is simple: it is enough to find the ratio of the distance traveled to the time during which it was overcome by the body. If the object moves in a circle, then in this case not a linear, but an angular velocity is already determined. What is this value and how is it calculated? This is exactly what will be discussed in this article.
Angular velocity: concept and formula
When a material point moves along a circle, the speed of its movement can be characterized by the value of the angle of rotation of the radius that connects the moving object with the center of this circle. It is clear that this value is constantly changing depending on time. The speed with which this process occurs is nothing but the angular velocity. In other words, this is the ratio of the magnitude of the deviation of the radiusvector of the object to the time interval that it took the object to make such a turn. The angular velocity formula (1) can be written as follows:
w =φ / t, where:
φ – radius rotation angle, t – rotation time period.
Units of measurement
In the international system of common units (SI), it is customary to use radians to characterize turns. Therefore, 1 rad/s is the basic unit used in angular velocity calculations. At the same time, no one forbids the use of degrees (recall that one radian is equal to 180 / pi, or 57˚18 '). Also, the angular velocity can be expressed in revolutions per minute or per second. If the movement along the circle occurs uniformly, then this value can be found by the formula (2):
w =2πn, where n is the speed.
Otherwise, just as it is done for normal speed, average or instantaneous angular speed is calculated. It should be noted that the quantity under consideration is a vector one. To determine its direction, the gimlet rule is usually used, which is often used in physics. The angular velocity vector is directed in the same direction as the translational movement of the screw with a right-hand thread. In other words, it is directed along the axis around which the body rotates, in the direction from which the rotation is seen to occur counterclockwise.
Calculation examples
Suppose you want to determine what the linear and angular speed of the wheel is, if it is known that its diameter is one meter, and the angle of rotation changes in accordance with the law φ=7t. Let's use our first formula:
w =φ / t=7t / t=7 s-1.
This will be the desired angular velocity. Now let's move on to finding the usual speed of movement. As you know, v=s / t. Given that s in our case is the circumference of the wheel (l=2πr), and 2π is one full turn, we get the following:
v=2πr / t=wr=70.5=3.5 m/s
Here's another problem on this topic. It is known that the radius of the Earth at the equator is 6370 kilometers. It is required to determine the linear and angular speed of movement of points located on this parallel, which occurs as a result of the rotation of our planet around its axis. In this case, we need the second formula:
w =2πn=23, 14 (1/(243600))=7, 268 10-5 rad/s.
It remains to find out what the linear speed is: v=wr=7, 268 10-5 63701000=463 m/s.
