MEAN ANGULAR VELOCITY: DEFINITION AND FORMULAS, SOLVED EXERCISES - PHYSICAL - 2024

The mean angular speed of rotation is defined as the angle rotated per unit time of the position vector of a point that describes circular motion. The blades of a ceiling fan (like the one shown in figure 1), follow circular motion and their average angular speed of rotation is calculated by taking the quotient between the angle rotated and the time in which that angle was traveled.

The rules that rotational motion follows are somewhat similar to the familiar ones for translational motion. The distances traveled can also be measured in meters, however the angular magnitudes are especially relevant because they greatly facilitate the description of the movement.

Figure 1. The fan blades have angular velocity. Source: Pixabay

In general, Greek letters are used for angular quantities and Latin letters for the corresponding linear quantities.

Definition and formulas

In figure 2 the movement of a point on a circular path c is represented. The position P of the point corresponds to the instant t and the angular position corresponding to that instant is ϕ.

From the instant t, a period of time Δt elapses. In that period the new position of the point is P 'and the angular position has increased by an angle Δϕ.

Figure 2. Circular motion of a point. Source: self made

The mean angular velocity ω is the angle traveled per unit of time, so the quotient Δϕ / Δt will represent the mean angular velocity between times t and t + Δt:

Since angle is measured in radians and time in seconds, the unit for mean angular velocity is rad / s. If we want to calculate the angular velocity just at the instant t, then we will have to calculate the ratio Δϕ / Δt when Δt ➡0.

Uniform rotation

A rotational movement is uniform if at any observed instant, the angle traveled is the same in the same period of time. If the rotation is uniform, then the angular velocity at any instant coincides with the mean angular velocity.

In a uniform rotational motion the time in which one complete revolution is made is called the period and is denoted by T.

In addition, when a complete turn is made, the angle traveled is 2π, so in a uniform rotation the angular velocity ω is related to the period T, by the following formula:

The frequency f of a uniform rotation is defined as the quotient between the number of turns and the time used to go through them, that is, if N turns are made in the time Δt then the frequency will be:

f = N / Δt

Since one turn (N = 1) is traveled in time T (the period), the following relationship is obtained:

f = 1 / T

That is, in a uniform rotation the angular velocity is related to the frequency through the relation:

ω = 2π ・ f

Relationship between angular speed and linear speed

The linear speed v, is the quotient between the distance traveled and the time taken to travel it. In figure 2 the distance traveled is the arc length Δs.

The arc Δs is proportional to the traveled angle Δϕ and the radius r, the following relationship being fulfilled:

Δs = r ・ Δϕ

Provided that Δϕ is measured in radians.

If we divide the previous expression by the time lapse Δt we will obtain:

(Δs / Δt) = r ・ (Δϕ / Δt)

The quotient of the first member is the linear speed and the quotient of the second member is the mean angular velocity:

v = r ・ ω

Solved exercises

-Exercise 1

The tips of the ceiling fan blades shown in figure 1 move with a speed of 5 m / s and the blades have a radius of 40 cm.

With these data, calculate: i) the average angular velocity of the wheel, ii) the number of turns the wheel makes in one second, iii) the period in seconds.

Solution

i) The linear speed is v = 5 m / s.

The radius is r = 0.40 m.

From the relationship between linear speed and angular speed we solve the latter:

v = r ・ ω => ω = v / r = (5 m / s) / (0.40 m) = 12.57 rad / s

ii) ω = 2π ・ f => f = ω / 2π = (12.57 rad / s) / (2π rad) = 2 turn / s

iii) T = 1 / f = 1 / (2 turn / s) = 0.5 s for each turn.

-Exercise 2

A toy stroller moves on a circular track with a radius of 2m. At 0s its angular position is 0 rad, but after time t its angular position is

φ (t) = 2 ・ t.

With this data

i) Calculate the mean angular velocity in the following time intervals;; and finally in the lapse.

ii) Based on the results of part i) What can be said about the movement?

iii) Determine the mean linear speed in the same period of time from part i)

iv) Find the angular velocity and linear speed for any instant.

Solution

i) The mean angular velocity is given by the following formula:

We proceed to calculate the angle traveled and the lapse of time elapsed in each interval.

Interval 1: Δϕ = ϕ (0.5s) - ϕ (0.0s) = 2 (rad / s) * 0.5s - 2 (rad / s) * 0.0s = 1.0 rad

Δt = 0.5s - 0.0s = 0.5s

ω = Δϕ / Δt = 1.0rad / 0.5s = 2.0 rad / s

Interval 2: Δϕ = ϕ (1.0s) - ϕ (0.5s) = 2 (rad / s) * 1.0s - 2 (rad / s) * 0.5s = 1.0 rad

Δt = 1.0s - 0.5s = 0.5s

ω = Δϕ / Δt = 1.0rad / 0.5s = 2.0 rad / s

Interval 3: Δϕ = ϕ (1.5s) - ϕ (1.0s) = 2 (rad / s) * 1.5s - 2 (rad / s) * 1.0s = 1.0 rad

Δt = 1.5s - 1.0s = 0.5s

ω = Δϕ / Δt = 1.0rad / 0.5s = 2.0 rad / s

Interval 4: Δϕ = ϕ (1.5s) - ϕ (0.0s) = 2 (rad / s) * 1.5s - 2 (rad / s) * 0.0s = 3.0 rad

Δt = 1.5s - 0.0s = 1.5s

ω = Δϕ / Δt = 3.0rad / 1.5s = 2.0 rad / s

ii) In view of the previous results, in which the average angular velocity was calculated in different time intervals, always obtaining the same result, it seems to indicate that it is a uniform circular motion. However, these results are not conclusive.

The way to ensure the conclusion is to calculate the mean angular velocity for an arbitrary interval: Δϕ = ϕ (t ') - ϕ (t) = 2 * t' - 2 * t = 2 * (t'-t)

Δt = t '- t

ω = Δϕ / Δt = 2 * (t'-t) / (t'-t) = 2.0 rad / s

This means that the toy stroller has a constant mean angular velocity of 2 rad / s in any period of time considered. But you can go further if you calculate the instantaneous angular velocity:

This is interpreted as that the toy car at all times has constant angular velocity = 2 rad / s.

References

1. Giancoli, D. Physics. Principles with Applications. 6th Edition. Prentice Hall. 30- 45.
2. Kirkpatrick, L. 2007. Physics: A Look at the World. 6 ^ta Editing abbreviated. Cengage Learning. 117.
3. Resnick, R. (1999). Physical. Volume 1. Third edition in Spanish. Mexico. Compañía Editorial Continental SA de CV 33-52.
4. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7th. Edition. Mexico. Cengage Learning Editors. 32-55.
5. Wikipedia. Angular velocity. Recovered from: wikipedia.com