THE angular speed is the velocity in circular paths. We can calculate this vector physical quantity by dividing the angular displacement by the time, in addition, we can find it through the hourly function of the position in the MCU and its relation to the period or the frequency.
Know more: Vector and Scalar Quantities—What's the Difference?
Summary on Angular Velocity
Angular velocity measures how fast the angular displacement occurs.
Whenever we have circular motions, we have angular velocity.
We can calculate velocity by dividing the angular displacement by time, the hourly function of the position in the MCU, and the relationship it has to period or frequency.
Period is the opposite of angular frequency.
The main difference between angular velocity and scalar velocity is that the former describes circular motions, while the latter describes linear motions.
What is Angular Velocity?
Angular velocity is a greatness vector physics describing motions around a circular path, measuring how quickly they happen.
Circular motion can be uniform, called uniform circular motion (MCU), which occurs when the angular velocity is constant and therefore the angular acceleration is zero. And it can also be uniform and varied, known as uniformly variable circular motion (MCUV), in which the angular velocity varies and we must consider the acceleration in the motion.
What are the formulas for angular velocity?
→ average angular velocity
\(\omega_m=\frac{∆φ}{∆t}\)
\(\omega_m\) → average angular velocity, measured in radiands per second \([rad/s]\).
\(∆φ\) → variation of angular displacement, measured in radians \([rad]\).
\(∆t\) → time variation, measured in seconds \([s]\).
Remembering that the displacement can be found using the following two formulas:
\(∆φ=φf-φi\)
\(∆φ=\frac{∆S}R\)
\(∆φ\) → variation of angular displacement or angle, measured in radians \([rad]\).
\(\varphi_f\) → final angular displacement, measured in radians \([rad]\).
\(\varphi_i\) → initial angular displacement, measured in radians \([rad]\).
\(∆S\) → variation of scalar displacement, measured in meters \([m]\).
R → radius of circumference.
In addition time variation can be calculated by the formula:
\(∆t=tf-ti\)
\(∆t\) → time variation, measured in seconds \([s]\).
\(t_f\) → final time, measured in seconds \([s]\).
\(you\) → start time, measured in seconds \([s]\).
→ Position time function in the MCU
\(\varphi_f=\varphi_i+\omega\bullet t\)
\(\varphi_f\) → final angular displacement, measured in radiands \(\left[rad\right]\).
\(\varphi_i\) → initial angular displacement, measured in radiands \([rad]\).
\(\omega\) → angular velocity, measured in radiands per second\(\left[{rad}/{s}\right]\).
t → time, measured in seconds [s].
How to calculate angular velocity?
We can find the average angular velocity by dividing the change in angular displacement by the change in time.
Example:
A wheel had an initial angular displacement of 20 radians and a final angular displacement of 30 radians during the time of 100 seconds, what was its average angular velocity?
Resolution:
Using the formula for average angular velocity, we will find the result:
\(\omega_m=\frac{∆φ}{∆t}\)
\(\omega_m=\frac{φf-φi}{∆t}\)
\(\omega_m=\frac{30-20}{100}\)
\(\omega_m=\frac{10}{100}\)
\(\omega_m=0.1\rad/s\)
The average speed of the wheel is 0.1 radian per second.
What is the relationship between angular velocity and period and frequency?
Angular velocity can be related to the period and frequency of motion. From the relationship between angular velocity and frequency, we get the formula:
\(\omega=2\bullet\pi\bullet f\)
\(\omega \) → angular velocity, measured in radiands per second \([rad/s]\).
\(f \) → frequency, measured in Hertz \([Hz]\).
Remembering that period is the opposite of frequency, as in the formula below:
\(T=\frac{1}{f}\)
\(T\) → period, measured in seconds \([s]\).
\(f\) → frequency, measured in Hertz \([Hz]\).
Based on this relationship between period and frequency, we were able to find the relationship between angular velocity and period, as in the formula below:
\(\omega=\frac{2\bullet\pi}{T}\)
\(\omega\) → angular velocity, measured in radiands per second \( [rad/s]\).
\(T \) → period, measured in seconds \(\left[s\right]\).
Difference between angular velocity and scalar velocity
Scalar or linear velocity measures how fast a linear motion occurs., being calculated by the linear displacement divided by the time. Unlike angular velocity, which measures how fast a circular motion occurs, being calculated by angular displacement divided by time.
We can relate the two by the formula:
\(\omega=\frac{v}{R}\)
\(\omega\) → is the angular velocity, measured in radiands per second \([rad/s]\).
\(v\) → is the linear speed, measured in meters per second \([m/s]\).
R → is the radius of the circle.
Read too: Average speed — a measure of how quickly the position of a piece of furniture changes
Solved exercises on angular velocity
question 1
The tachometer is a piece of equipment that is located on the dashboard of the car to indicate to the driver in real time what the engine rotation frequency is. Assuming that a tachometer is indicating 3000 rpm, determine the angular speed of rotation of the engine in rad/s.
A) 80 π
B) 90 π
C) 100 π
D) 150 π
E) 200 π
Resolution:
Alternative C
The angular speed of rotation of the motor is calculated by the formula:
\(\omega=2\bullet\pi\bullet f\)
Since the frequency is in rpm (revolutions per minute), we have to convert it to Hz, dividing rpm by 60 minutes:
\(\frac{3000\ revolutions}{60\ minutes}=50 Hz\)
Substituting into the angular velocity formula, then its value is:
\(\omega=2\bullet\pi\bullet50\)
\(\omega=100\pi\rad/s\)
question 2
(UFPR) A point in uniform circular motion describes 15 revolutions per second in a circle with a radius of 8.0 cm. Its angular velocity, period and linear velocity are, respectively:
A) 20 rad/s; (1/15) s; 280 π cm/s.
B) 30 rad/s; (1/10) s; 160 π cm/s.
C) 30 π rad/s; (1/15) s; 240 π cm/s.
D) 60 π rad/s; 15 s; 240 π cm/s.
E) 40 π rad/s; 15 s; 200 π cm/s.
Resolution:
Alternative C
Knowing that the frequency is 15 revolutions per second or 15 Hz, then the angular velocity is:
\(\omega=2\bullet\pi\bullet f\)
\(\omega=2\bullet\pi\bullet15\)
\(\omega=30\pi\rad/s\)
The period is the inverse of the frequency, so:
\(T=\frac{1}{f}\)
\(T=\frac{1}{15}\ s\)
Finally, the linear velocity is:
\(v=\omega\bullet r\)
\(v=30\pi\bullet8\)
\(v=240\pi\ cm/s\)
By Pâmella Raphaella Melo
Physics teacher
Source: Brazil School - https://brasilescola.uol.com.br/fisica/velocidade-angular.htm