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Constant angular acceleration
Storyboard 
To achieve a certain angular velocity, an object must first increase its angular velocity from rest. This process is called angular acceleration and is defined in terms of the change in angular velocity over time. On the other hand, if the goal is to decrease the angular velocity and even stop the rotation of the object, angular deceleration is introduced, with the opposite sign to that of the angular velocity (if the angular velocity is positive, the angular deceleration is negative, and vice versa), known as angular braking.
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Mechanisms
Iframe
Mechanisms
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Mean angular acceleration
Concept
When angular velocity isn't constant, it's crucial to understand how it changes over time. To do this, we need to know the rate of change of angular velocity per unit of time, known as angular acceleration or deceleration, depending on whether the angular velocity is increasing or decreasing.
Angular acceleration is determined by measuring the variation of angular velocity over time.
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Measuring mean angular acceleration
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The average angular acceleration is defined as the proportion in which the angular velocity changes over time. To measure this quantity accurately, it is necessary to quantify how the angular velocity changes over the course of time.
The equation that describes this average angular acceleration is as follows:
| $ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$ |
It is important to note that the average angular acceleration is an estimation of the actual angular acceleration. However, there is a fundamental issue:
If the angular acceleration varies over time, the value of the average angular acceleration can differ significantly from the average angular acceleration.
Therefore, the key lies in
Determining the angular acceleration within a sufficiently short time interval to minimize any significant variation.
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Angular speed in the case of constant angular acceleration
Description
In the case of constant angular acceleration, the angular velocity follows a linear relationship with respect to time:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
which is depicted in the following graph:
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Angle traveled for constant angular acceleration
Concept
With the constant Acceleration ($a_0$), the function of the angular Speed ($\omega$) describes a line whose slope is equal to the angular acceleration. Along with the initial Angular Speed ($\omega_i$), the time ($t$), and the start Time ($t_0$), the relationship is expressed by the equation:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
Therefore, the area under a curve, which represents the total displacement, consists of a rectangle and a triangle:
The rectangle has a height corresponding to the initial velocity and a base equal to the elapsed time. The triangle, on the other hand, has a height that is the product of the angular acceleration times the elapsed time, and a base that is also equal to the elapsed time. With this information, the total displacement the angle ($\theta$) can be calculated using the initial Angle ($\theta_0$) as shown below:
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$ |
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Tangential acceleration, right hand rule
Image
The orientation of tangential acceleration can be obtained using the right-hand rule, with fingers pointing towards the axis and then rotating towards the radius:
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Model
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Calculations
Variables
Parameters
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Calculations
Calculations
Variable 
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Target : Equation To be used Equation
$ a_0 = r \alpha_0 $
a = r * alpha
$ \bar{\alpha} = \alpha_0 $
alpha_m = alpha_0
$ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$
alpha_m = Domega / Dt
$ \Delta\omega = \omega - \omega_0 $
Domega = omega - omega_0
$ \Delta t \equiv t - t_0 $
Dt = t - t_0
$ \Delta\theta = \theta - \theta_0 $
Dtheta = theta - theta_0
$ \omega = \omega_0 + \alpha_0 ( t - t_0 )$
omega = omega_0 + alpha_0 * ( t - t_0 )
$ \theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$
theta = theta_0 + omega_0 *( t - t_0 )+ alpha_0 *( t - t_0 )^2/2
$ \theta = \theta_0 +\displaystyle\frac{ \omega ^2- \omega_0 ^2}{2 \alpha_0 }$
theta = theta_0 +( omega ^2 - omega_0 ^2)/(2* alpha_0 )
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Mean Angular Acceleration
Equation
The rate at which angular velocity changes over time is defined as the mean Angular Acceleration ($\bar{\alpha}$). To measure it, we need to observe the difference in Angular Speeds ($\Delta\omega$) and the time elapsed ($\Delta t$).
The equation describing the mean Angular Acceleration ($\bar{\alpha}$) is as follows:
| $ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$ |
The definition of average angular acceleration is based on the angle covered
| $ \Delta\omega = \omega - \omega_0 $ |
and the elapsed time
| $ \Delta t \equiv t - t_0 $ |
The relationship between the two is defined as the average angular acceleration
| $ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$ |
within that time interval.
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Constant angular acceleration
Equation
If the acceleration does not vary, the mean Angular Acceleration ($\bar{\alpha}$) will be equal to the constant Angular Acceleration ($\alpha_0$), which is expressed as:
| $ \bar{\alpha} = \alpha_0 $ |
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Angle Difference
Equation
To describe the rotation of an object, we need to determine the angle variation ($\Delta\theta$). This is achieved by subtracting the initial Angle ($\theta_0$) from the angle ($\theta$), which is reached by the object during its rotation:
| $ \Delta\theta = \theta - \theta_0 $ |
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Variation of angular speeds
Equation
Acceleration is defined as the change in angular velocity per unit of time.
Therefore, the angular acceleration the difference in Angular Speeds ($\Delta\omega$) can be expressed in terms of the angular velocity the angular Speed ($\omega$) and time the initial Angular Speed ($\omega_i$) as follows:
| $ \Delta\omega = \omega - \omega_0 $ |
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Elapsed time
Equation
To describe the motion of an object, we need to calculate the time elapsed ($\Delta t$). This magnitude is obtained by measuring the start Time ($t_0$) and the the time ($t$) of said motion. The duration is determined by subtracting the initial time from the final time:
| $ \Delta t \equiv t - t_0 $ |
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Angular velocity with constant angular acceleration
Equation
With the constant Angular Acceleration ($\alpha_0$), the angular Speed ($\omega$) forms a linear relationship with the time ($t$), incorporating the variables the initial Angular Speed ($\omega_i$) and the start Time ($t_0$) as follows:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
If we assume that the mean Angular Acceleration ($\bar{\alpha}$) is constant, equivalent to the constant Angular Acceleration ($\alpha_0$), then the following equation applies:
| $ \bar{\alpha} = \alpha_0 $ |
Therefore, considering the difference in Angular Speeds ($\Delta\omega$) along with the angular Speed ($\omega$) and the initial Angular Speed ($\omega_i$):
| $ \Delta\omega = \omega - \omega_0 $ |
and the time elapsed ($\Delta t$) in relation to the time ($t$) and the start Time ($t_0$):
| $ \Delta t \equiv t - t_0 $ |
the equation for the mean Angular Acceleration ($\bar{\alpha}$):
| $ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$ |
can be expressed as:
$\alpha_0 = \alpha = \displaystyle\frac{\Delta \omega}{\Delta t} = \displaystyle\frac{\omega - \omega_0}{t - t_0}$
Solving this, we obtain:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
This equation represents a straight line in the angular velocity versus time plane.
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Angle at Constant Angular Acceleration
Equation
Given that the total displacement corresponds to the area under the angular velocity versus time curve, in the case of a constant Angular Acceleration ($\alpha_0$), it is determined that the displacement the angle ($\theta$) with the variables the initial Angle ($\theta_0$), the time ($t$), the start Time ($t_0$), and the initial Angular Speed ($\omega_i$) is as follows:
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$ |
In the case of the constant Angular Acceleration ($\alpha_0$), the angular Speed ($\omega$) as a function of the time ($t$) follows a linear relationship with the start Time ($t_0$) and the initial Angular Speed ($\omega_i$) in the form of:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
Given that the angular displacement is equal to the area under the angular velocity-time curve, in this case, one can add the contributions of the rectangle:
$\omega_0(t-t_0)$
and the triangle:
$\displaystyle\frac{1}{2}\alpha_0(t-t_0)^2$
This leads us to the expression for the angle ($\theta$) and the initial Angle ($\theta_0$):
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$ |
This expression corresponds to the general form of a parabola.
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Braking angle as a function of angular velocity
Equation
In the case of the constant Angular Acceleration ($\alpha_0$), the function of the angular Speed ($\omega$) with respect to the time ($t$), along with additional variables the initial Angular Speed ($\omega_i$) and the start Time ($t_0$), is expressed by the equation:
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
From this equation, it is possible to calculate the relationship between the angle ($\theta$) and the initial Angle ($\theta_0$), as well as the change in angular velocity:
| $ \theta = \theta_0 +\displaystyle\frac{ \omega ^2- \omega_0 ^2}{2 \alpha_0 }$ |
If we solve for time in the equation of the angular Speed ($\omega$) that includes the variables the initial Angular Speed ($\omega_i$), the time ($t$), the start Time ($t_0$), and the constant Angular Acceleration ($\alpha_0$):
| $ \omega = \omega_0 + \alpha_0 ( t - t_0 )$ |
we obtain the following expression for time:
$t - t_0 = \displaystyle\frac{\omega - \omega_0}{\alpha_0}$
This solution can be substituted into the equation to calculate the angle ($\theta$) using the initial Angle ($\theta_0$) as follows:
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$ |
which results in the following equation:
| $ \theta = \theta_0 +\displaystyle\frac{ \omega ^2- \omega_0 ^2}{2 \alpha_0 }$ |
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Acceleration and Angular Acceleration
Equation
If we divide the relationship between the mean Speed ($\bar{v}$), the radio ($r$), and the mean angular velocity ($\bar{\omega}$), expressed in the following equation:
| $ v = r \omega $ |
by the value of the time elapsed ($\Delta t$), we can obtain the factor that allows us to calculate the angular acceleration along the orbit:
| $ a_0 = r \alpha_0 $ |
| $ a = r \alpha $ |
Given that the mean Acceleration ($\bar{a}$) equals the speed Diference ($\Delta v$) and the time elapsed ($\Delta t$) according to
| $ \bar{a} \equiv\displaystyle\frac{ \Delta v }{ \Delta t }$ |
and the mean Angular Acceleration ($\bar{\alpha}$) equals the difference in Angular Speeds ($\Delta\omega$) and the time elapsed ($\Delta t$) as per
| $ \bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$ |
it follows that
$\bar{a}=\displaystyle\frac{\Delta v}{\Delta t}=r\displaystyle\frac{\Delta\omega}{\Delta t}=\bar{\alpha}$
Assuming that the mean Angular Acceleration ($\bar{\alpha}$) is equal to the constant Angular Acceleration ($\alpha_0$)
| $ \bar{\alpha} = \alpha_0 $ |
and assuming that the mean Acceleration ($\bar{a}$) equals the constant Acceleration ($a_0$)
| $ a_0 = \bar{a} $ |
then the following equation is obtained:
| $ a = r \alpha $ |
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