Usually, when talking about movement, we imagine an object that moves in a straight line. The speed of such a movement is usually called linear, and the calculation of its average value is simple: just find the ratio of the distance traveled to the time for which it was overcome by the body. If the object moves in a circle, then in this case the angular velocity rather than linear is already determined . What is this value and how is it calculated? This will be discussed in this article.
Angular velocity: concept and formula
When a material point moves in a circle, the speed of its movement can be characterized by the angle of rotation of the radius that connects the moving object with the center of the circle. It is clear that this value is constantly changing depending on time. The speed with which this process occurs is nothing but angular velocity. In other words, this is the ratio of the deviation of the radius vector of the object to the period of time that the object took to complete such a turn. The angular velocity formula (1) can be written in the following form:
w = φ / t, where:
φ is the angle of rotation of the radius,
t is the rotation time period.
Units of measure
In the international system of generally accepted units (SI), it is customary to use radians to characterize turns. Therefore, 1 rad / s is the basic unit that is used in calculating angular velocity. At the same time, no one forbids using degrees (recall that one radian is equal to 180 / pi, or 57˚18 '). Also, angular velocity can be expressed in revolutions per minute or per second. If the movement around the circle occurs uniformly, then this value can be found by the formula (2):
w = 2π * n,
where n is the rotation frequency.
Otherwise, just as they do for ordinary speed, calculate the average or instantaneous angular velocity. It should be noted that the quantity in question is vectorial. To determine its direction, the rule of the gimlet is usually used , which is often used in physics. The angular velocity vector is directed in the same direction in which the translational movement of the screw with the right thread occurs. In other words, it is directed along the axis around which the body rotates, in the direction from which the rotation is seen happening counterclockwise.
Calculation Examples
Suppose you want to determine what the linear and angular velocity of the wheel is, if it is known that its diameter is one meter, and the rotation angle changes in accordance with the law φ = 7t. We use our first formula:
w = φ / t = 7t / t = 7 s -1 .
This will be the desired angular velocity. Now we turn to the search for the usual speed of movement. As is known, v = s / t. Given that s in our case is the wheel circumference (l = 2π * r), and 2π is one full revolution, the following is obtained:
v = 2π * r / t = w * r = 7 * 0.5 = 3.5 m / s
Here is another puzzle 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 velocity of the 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 = 2π * n = 2 * 3.14 * (1 / (24 * 3600)) = 7.268 * 10 -5 rad / s.
It remains to find out what the linear velocity is: v = w * r = 7.268 * 10 -5 * 6370 * 1000 = 463 m / s.