What is the angular velocity and how is it calculated?

Usually when we talk about moving, we imagine an object that moves in a straight line. The speed of such motion is usually called linear, and the calculation of its average value is simple: it is sufficient to find the ratio of the distance traveled to the time for which it was overcome by the body. If the object moves along a circle, then in this case it is not the linear velocity but the angular velocity that is determined . What is this value and how does it count? 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 a given circle. It is clear that this quantity varies with time. The speed with which this process takes place is just the angular velocity. In other words, this is the ratio of the deviation of the radius-vector of the object to the time interval required for the object to make 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 period of time of rotation.

Units of measurement

In the international system of conventional units (SI), it is customary to use radians to characterize rotations. Therefore, 1 rad / s is the basic unit used in calculating the angular velocity. At the same time, no one forbids the use of degrees (recall that one radian is 180 / pi, or 57˚18 '). Also, the angular velocity can be expressed in revolutions per minute or per second. If the displacement along the circumference occurs uniformly, then this value can be found by the formula (2):

W = 2π * n,

Where n is the rotational speed.

Otherwise, just like doing it for normal speed, calculate the average, or instantaneous angular velocity. It should be noted that the quantity in question is a vector quantity. To determine its direction, a rule is usually used , which is often used in physics. The angular velocity vector is directed in the same direction as the right-handed screw. In other words, it is directed along the axis around which the body rotates, in the direction from which the rotation is seen going against the clockwise movement.

Calculation examples

Suppose you want to determine what the linear and angular velocity of a wheel is equal to if it is known that its diameter is one meter and the rotation angle changes according to the law φ = 7t. We use our first formula:

W = φ / t = 7t / t = 7 s ^-1 .

This is the required angular velocity. Now let's move on to finding the speed of movement that we are familiar with. As is well known, v = s / t. Considering that s in our case is the circumference of the wheel (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's another puzzle on this topic. It is known that the Earth's radius at the equator is 6370 kilometers. It is required to determine the linear and angular velocity of the points located on this parallel, which arises 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 equal to: v = w * r = 7.268 * 10 ^-5 * 6370 * 1000 = 463 m / s.