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Constant angular acceleration, two stages
Storyboard 
In the case of an accelerated angular motion in two stages, when transitioning from the first to the second angular acceleration, the final angular velocity of the first stage becomes the initial angular velocity of the second stage. The same applies to the angle, where the final angle of the first stage equals the initial angle of the second stage.
Unlike the two-angular-velocity model, this model doesn't exhibit discontinuity issues, except that the angular acceleration can change abruptly, which is technically possible but often not very realistic.
ID:(1409, 0)
Two-stage movement
Description
In a two-stage motion scenario, initially the object adjusts its speed by the difference of the variation of angular velocities in the first stage ($\Delta\omega_1$) over a period of the time spent in the first stage ($\Delta t_1$), experiencing an acceleration of the angular acceleration during the first stage ($\alpha_1$).
| $ \alpha_1 \equiv \displaystyle\frac{ \Delta\omega_1 }{ \Delta t_1 }$ |
In the second stage, the object continues to modify its speed by the variation of angular velocities in the second stage ($\Delta\omega_2$) over a time span of the time spent in the second stage ($\Delta t_2$), with an acceleration of the angular acceleration during the second stage ($\alpha_2$).
| $ \alpha_2 \equiv \displaystyle\frac{ \Delta\omega_2 }{ \Delta t_2 }$ |
When visualized graphically, this results in a velocity versus time diagram as shown below:
It is important to note that the time intervals the time spent in the first stage ($\Delta t_1$) and the time spent in the second stage ($\Delta t_2$) are sequential, just like the speed differences the variation of angular velocities in the first stage ($\Delta\omega_1$) and the variation of angular velocities in the second stage ($\Delta\omega_2$).
ID:(12521, 0)
Angular velocity in a two-stage movement
Description
In the analysis of motion segmented into two stages, the first phase is characterized by a linear function that incorporates points the start Time ($t_0$), the final time of first and start of second stage ($t_1$), the initial Angular Speed ($\omega_0$), and the first final angular velocity and second stage start ($\omega_1$). This is expressed through a line with a slope of the angular acceleration during the first stage ($\alpha_1$), whose mathematical relationship is specified in the following equation:
| $ \omega_1 = \omega_0 + \alpha_1 ( t_1 - t_0 )$ |
Transitioning to the second stage, which is defined by the points the first final angular velocity and second stage start ($\omega_1$), the final angular velocity of the second stage ($\omega_2$), the final time of first and start of second stage ($t_1$), and the second stage ending time ($t_2$), a new linear function with a slope of the angular acceleration during the second stage ($\alpha_2$) is adopted. This relationship is outlined by the second equation presented:
| $ \omega_2 = \omega_1 + \alpha_2 ( t_2 - t_1 )$ |
The graphical representation of these linear relationships is illustrated below, providing a clear visualization of how the slope varies between the two stages:
ID:(12522, 0)
Angle in a two-stage movement
Description
In a scenario of motion divided into two stages, the angle at the end of the first stage is the same as the angle at the beginning of the second stage, designated as the first final angle and second stage began ($\theta_1$).
Similarly, the time at which the first stage ends coincides with the start of the second stage, marked by the final time of first and start of second stage ($t_1$).
Given that the motion is defined by the angular acceleration experienced, the angular velocity at the end of the first stage must match the initial angular velocity of the second stage, indicated by the first final angular velocity and second stage start ($\omega_1$).
In the context of constant angular acceleration, the angle at the first final angle and second stage began ($\theta_1$) is determined by the variables the initial Angle ($\theta_0$), the initial Angular Speed ($\omega_0$), the angular acceleration during the first stage ($\alpha_1$), the final time of first and start of second stage ($t_1$), and the start Time ($t_0$), as shown in the following equation:
| $ \theta_1 = \theta_0 + \omega_0 ( t_1 - t_0 )+\displaystyle\frac{1}{2} \alpha_1 ( t_1 - t_0 )^2$ |
In the second stage, the angle at the second stage final angle ($\theta_2$) is calculated based on the first final angle and second stage began ($\theta_1$), the first final angular velocity and second stage start ($\omega_1$), the angular acceleration during the second stage ($\alpha_2$), the final time of first and start of second stage ($t_1$), and the second stage ending time ($t_2$), according to:
| $ \theta_2 = \theta_1 + \omega_1 ( t_2 - t_1 )+\displaystyle\frac{1}{2} \alpha_2 ( t_2 - t_1 )^2$ |
The graphical representation of these relationships is illustrated below:
ID:(12520, 0)
Model
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Target : Equation To be used Equation
$ a_1 = r \alpha$
a = r * alpha
$ a_2 = r \alpha$
a = r * alpha
$ \alpha_1 \equiv \displaystyle\frac{ \Delta\omega_1 }{ \Delta t_1 }$
alpha_m = Domega / Dt
$ \alpha_2 \equiv \displaystyle\frac{ \Delta\omega_2 }{ \Delta t_2 }$
alpha_m = Domega / Dt
$ \Delta\omega_1 = \omega_1 - \omega_0 $
Domega = omega - omega_0
$ \Delta\omega_2 = \omega_2 - \omega_1 $
Domega = omega - omega_0
$ \Delta t_1 \equiv t_1 - t_0 $
Dt = t - t_0
$ \Delta t_2 \equiv t_2 - t_1 $
Dt = t - t_0
$ \Delta\theta_1 = \theta_1 - \theta_0 $
Dtheta = theta - theta_0
$ \Delta\theta_2 = \theta_2 - \theta_1 $
Dtheta = theta - theta_0
$ \omega_1 = \omega_0 + \alpha_1 ( t_1 - t_0 )$
omega = omega_0 + alpha_0 * ( t - t_0 )
$ \omega_2 = \omega_1 + \alpha_2 ( t_2 - t_1 )$
omega = omega_0 + alpha_0 * ( t - t_0 )
$ \theta_1 = \theta_0 + \omega_0 ( t_1 - t_0 )+\displaystyle\frac{1}{2} \alpha_1 ( t_1 - t_0 )^2$
theta = theta_0 + omega_0 *( t - t_0 )+ alpha_0 *( t - t_0 )^2/2
$ \theta_2 = \theta_1 + \omega_1 ( t_2 - t_1 )+\displaystyle\frac{1}{2} \alpha_2 ( t_2 - t_1 )^2$
theta = theta_0 + omega_0 *( t - t_0 )+ alpha_0 *( t - t_0 )^2/2
$ \theta_1 = \theta_0 +\displaystyle\frac{ \omega_1 ^2- \omega_0 ^2}{2 \alpha_1 }$
theta = theta_0 +( omega ^2 - omega_0 ^2)/(2* alpha_0 )
$ \theta_2 = \theta_1 +\displaystyle\frac{ \omega_2 ^2- \omega_1 ^2}{2 \alpha_2 }$
theta = theta_0 +( omega ^2 - omega_0 ^2)/(2* alpha_0 )
ID:(15424, 0)
Mean Angular Acceleration (1)
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:
| $ \alpha_1 \equiv \displaystyle\frac{ \Delta\omega_1 }{ \Delta t_1 }$ |
| $ \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.
ID:(3234, 1)
Mean Angular Acceleration (2)
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:
| $ \alpha_2 \equiv \displaystyle\frac{ \Delta\omega_2 }{ \Delta t_2 }$ |
| $ \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.
ID:(3234, 2)
Angle Difference (1)
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_1 = \theta_1 - \theta_0 $ |
| $ \Delta\theta = \theta - \theta_0 $ |
ID:(3680, 1)
Angle Difference (2)
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_2 = \theta_2 - \theta_1 $ |
| $ \Delta\theta = \theta - \theta_0 $ |
ID:(3680, 2)
Variation of angular speeds (1)
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 ($\omega_0$) as follows:
| $ \Delta\omega_1 = \omega_1 - \omega_0 $ |
| $ \Delta\omega = \omega - \omega_0 $ |
ID:(3681, 1)
Variation of angular speeds (2)
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 ($\omega_0$) as follows:
| $ \Delta\omega_2 = \omega_2 - \omega_1 $ |
| $ \Delta\omega = \omega - \omega_0 $ |
ID:(3681, 2)
Elapsed time (1)
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_1 \equiv t_1 - t_0 $ |
| $ \Delta t \equiv t - t_0 $ |
ID:(4353, 1)
Elapsed time (2)
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_2 \equiv t_2 - t_1 $ |
| $ \Delta t \equiv t - t_0 $ |
ID:(4353, 2)
Angular velocity with constant angular acceleration (1)
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 ($\omega_0$) and the start Time ($t_0$) as follows:
| $ \omega_1 = \omega_0 + \alpha_1 ( t_1 - t_0 )$ |
| $ \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 ($\omega_0$):
| $ \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.
ID:(3237, 1)
Angular velocity with constant angular acceleration (2)
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 ($\omega_0$) and the start Time ($t_0$) as follows:
| $ \omega_2 = \omega_1 + \alpha_2 ( t_2 - t_1 )$ |
| $ \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 ($\omega_0$):
| $ \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.
ID:(3237, 2)
Angle at Constant Angular Acceleration (1)
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 ($\omega_0$) is as follows:
| $ \theta_1 = \theta_0 + \omega_0 ( t_1 - t_0 )+\displaystyle\frac{1}{2} \alpha_1 ( t_1 - t_0 )^2$ |
| $ \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 ($\omega_0$) 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.
ID:(3682, 1)
Angle at Constant Angular Acceleration (2)
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 ($\omega_0$) is as follows:
| $ \theta_2 = \theta_1 + \omega_1 ( t_2 - t_1 )+\displaystyle\frac{1}{2} \alpha_2 ( t_2 - t_1 )^2$ |
| $ \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 ($\omega_0$) 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.
ID:(3682, 2)
Braking angle as a function of angular velocity (1)
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 ($\omega_0$) 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_1 = \theta_0 +\displaystyle\frac{ \omega_1 ^2- \omega_0 ^2}{2 \alpha_1 }$ |
| $ \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 ($\omega_0$), 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 }$ |
ID:(4386, 1)
Braking angle as a function of angular velocity (2)
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 ($\omega_0$) 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_2 = \theta_1 +\displaystyle\frac{ \omega_2 ^2- \omega_1 ^2}{2 \alpha_2 }$ |
| $ \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 ($\omega_0$), 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 }$ |
ID:(4386, 2)
Acceleration and Angular Acceleration (1)
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_1 = r \alpha$ |
| $ 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$ |
ID:(3236, 1)
Acceleration and Angular Acceleration (2)
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_2 = r \alpha$ |
| $ 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$ |
ID:(3236, 2)