THE angular acceleration is the measure of the angular velocity necessary for, in a specific time, a path to be covered. We can calculate it by dividing the variation of angular velocity with time and also by the time functions of angular position and angular velocity.
Read too: After all, what is acceleration?
Summary on Angular Acceleration
- When the angular velocity varies, there is considerable angular acceleration.
- In uniform circular motion, angular acceleration is zero, but in uniformly varied circular motion, there is angular acceleration.
- Angular acceleration occurs in circular paths; linear acceleration, in rectilinear paths.
- Torricelli's equation, used in linear motion, can also be employed in circular motion.
What is angular acceleration?
Angular acceleration is a vector physical quantity that describes the angular velocity in a circular path during a time interval.
When we consider the motion as uniform, that is, with constant angular velocity, we have zero angular acceleration, as in the case of uniform circular motion (
MCU). But if we consider the motion to occur in a uniformly varied way, the angular velocity varies. Thus, angular acceleration becomes indispensable in calculations, as in the case of uniformly variable circular motion (MCUV).Angular Acceleration Formula
average angular acceleration
\(\alpha_m=\frac{∆ω}{∆t}\)
⇒ αm is the average angular acceleration, measured in [rad/s2].
⇒ ∆ω is the change in angular velocity, measured in [rad/s].
⇒ ∆t is the change in time, measured in seconds [s].
Speed time function in MCUV
\(\omega_f=\omega_i+\alpha\bullet t\)
⇒ ωf is the final angular velocity, measured in [rad/s].
⇒ ωi is the initial angular velocity, measured in [rad/s].
⇒ α is the angular acceleration, measured in [rad/s2].
⇒ t is time, measured in seconds [s].
Position time function in the MCUV
\(\varphi_f=\varphi_i+\omega_i\bullet t+\frac{\alpha\bullet t^2}{2}\)
⇒ φf is the final angular displacement, measured in radians [rad].
⇒ φi is the initial angular displacement, measured in radians [rad].
⇒ ωi is the initial angular velocity, measured in [rad/s].
⇒ α is the angular acceleration, measured in [rad/s2].
⇒ t is time, measured in seconds [s].
How is angular acceleration calculated?
We can calculate angular acceleration using their formulas. To better understand how this works, we will see some examples below.
Example 1: If a wheel with an angular speed of 0,5rad/s rotate for 1.25 seconds, what is its average angular acceleration?
Resolution
We will find the angular acceleration by the formula:
\(\alpha_m=∆ωt\)
\(\alpha_m=\frac{0.5}{1.25}\)
\(\alpha_m=0.4{rad}/{s^2}\)
The average acceleration is \(0.4{rad}/{s^2}\).
Example 2: An individual set out on a bicycle and took 20 seconds to reach his destination. Knowing that the final angular displacement of the wheel was 100 radians, what was its acceleration?
Resolution:
Since it started from rest, its initial angular velocity and displacement are zero. We will find the acceleration using the formula for the hourly function of the position in the MCU:
\(\varphi_f=\varphi_i+\omega_i\bullet t+\frac{\alpha\bullet t^2}{2}\)
\(100=0+0\bullet20+\frac{\alpha\bullet{20}^2}{2}\)
\(100=20+\frac{\alpha\bullet400}{2}\)
\(100-20=\frac{\alpha\bullet400}{2}\)
\(80=\alpha\bullet200\)
\(\frac{80}{200}=\alpha\)
\(\alpha=0.4{rad}/{s^2}\)
Acceleration is valid \(0.4{rad}/{s^2}\).
Read too: Centripetal acceleration — that which is present in all circular motions
Differences between angular acceleration and linear acceleration
THE scalar or linear acceleration happens when there is a linear motion, being calculated by means of the linear velocity divided by the time. Angular acceleration appears in circular motions and can be found through angular velocity divided by time.
Angular and linear accelerations are related through the formula:
\(\alpha=\frac{a}{R}\)
- α is the angular velocity, measured in [rad/s2].
- The is the linear acceleration, measured in [m/s2].
- R is the radius of the circle.
Torricelli's equation
THE Torricelli's equation, used for linear movements, can also be used for circular movements, if the representation and meaning of the variables are changed. In this way, the equation can be rewritten as follows:
\(\omega_f^2=\omega_0^2+2\bullet\alpha\bullet∆φ\)
- ωf is the final angular velocity, measured in radians per second [rad/s].
- ω0is the initial angular velocity, measured in radians per second [rad/s].
- α is the angular acceleration, measured in [rads/2].
- ∆φ is the change in angular displacement, measured in radians [rad].
Solved exercises on angular acceleration
question 1
A centrifuge has a maximum spin speed of 30 radians per second, which is reached after 10 complete revolutions. What is your average acceleration? Use π = 3.
a) 12
b) 20
c) 7.5
d) 6
e) 10
Resolution:
Alternative C
First, we will find the value of the angular displacement by means of a simple rule of three:
\(1turn-2\bullet\pi rad\)
\(10 laps-∆φ\)
\(∆φ=10∙2∙πrad\)
\(∆φ=20∙πrad\)
To calculate the angular acceleration in this case, we will use Torricelli's formula:
\(\omega_f^2=\omega_0^2+2\bullet\alpha\bullet∆φ\)
The maximum speed corresponds to the final angular speed, which is 60. Therefore, the initial angular velocity was 0:
\({30}^2=0^2+2\bullet\alpha\bullet20\bullet\pi\)
\(900=0+\alpha\bullet40\bullet\pi\)
\(900=\alpha\bullet40\bullet3\)
\(900=\alpha\bullet120\)
\(\frac{900}{120}=\alpha\)
\(7.5{rad}/{s^2}=\alpha\)
question 2
A particle has an angular acceleration that varies with time, according to the equation\(\alpha=6t+3t^2\). Find the angular velocity and angular acceleration at the instant \(t=2s\).
Resolution:
At first, we will find the angular acceleration at the instant \(t=2s\), Substituting its value in the equation:
\(\alpha=6t+3t^2\)
\(\alpha=6\bullet2+3{\bullet2}^2\)
\(\alpha=12+12\)
\(\alpha=24{rad}/{s^2}\)
The angular velocity at the instant \(t=2s\) can be found using the formula for the average acceleration:
\(\alpha_m=∆ω∆t\)
\(24=\frac{\omega}{2}\)
\(\omega=2\bullet24\)
\(\omega=48 {rad}/{s}\)
By Pâmella Raphaella Melo
Physics teacher
Source: Brazil School - https://brasilescola.uol.com.br/fisica/aceleracao-angular.htm
