Storyboard

The objects can intersect when they coincide in angle at the same instant. To achieve this, they must move from their respective initial angles and angular velocities with angular accelerations that allow them to coincide in angle and time at the end of the journey.

>Model

ID:(1451, 0)

Iframe

>Top

ID:(15416, 0)

Concept

>Top

In a scenario of two-body motion, the first one alters the angular velocity difference of the first body ($\Delta\omega_1$) during the travel time of first object ($\Delta t_1$) with the angular acceleration of the first body ($\alpha_1$).

Subsequently, the second body advances, altering the angular velocity difference of the second body ($\Delta\omega_2$) during the travel time of second object ($\Delta t_2$) with the angular acceleration of the second body ($\alpha_2$).

Graphically represented, we obtain a velocity-time diagram as shown below:

The key here is that the values the angular velocity difference of the first body ($\Delta\omega_1$) and the angular velocity difference of the second body ($\Delta\omega_2$), and the values the travel time of first object ($\Delta t_1$) and the travel time of second object ($\Delta t_2$), are such that both bodies coincide in angle and time.

ID:(10579, 0)

Concept

>Top

In the case of two bodies, the motion of the first can be described by a function involving the points the initial angular velocity of the first body ($\omega_{01}$), the final angular velocity of the first body ($\omega_1$), the intersection time ($t$), and the initial time of first object ($t_1$), represented by a line with a slope of the angular acceleration of the first body ($\alpha_1$):

For the motion of the second body, defined by the points the initial angular velocity of the second body ($\omega_{02}$), the final angular velocity of the second body ($\omega_2$), the initial time of second object ($t_2$), and the intersection time ($t$), a second line with a slope of the angular acceleration of the second body ($\alpha_2$) is utilized:

This is represented as:

ID:(9872, 0)

Description

>Top

In the case of a two-body motion, the angle at which the trajectory of the first ends coincides with that of the second body at the angle of intersection ($\theta$).

Similarly, the time at which the trajectory of the first ends coincides with that of the second body at the intersection time ($t$).

For the first body, the angle of intersection ($\theta$) depends on the initial angle of the first body ($\theta_1$), the initial angular velocity of the first body ($\omega_{01}$), the angular acceleration of the first body ($\alpha_1$), the initial time of first object ($t_1$), as follows:

While for the second body, the angle of intersection ($\theta$) depends on the initial angle of the second body ($\theta_2$), the initial angular velocity of the second body ($\omega_{02}$), the angular acceleration of the second body ($\alpha_2$), the initial time of second object ($t_2$), as follows:

This is represented as:

ID:(12514, 0)

Top

>Top

Parameters

$\alpha_1$

alpha_1

Angular acceleration of the first body

rad/s^2

$\alpha_2$

alpha_2

Angular acceleration of the second body

rad/s^2

$a_1$

a_1

First body acceleration

m/s^2

$r$

r

Radio

m

$a_2$

a_2

Second body acceleration

m/s^2

Variables

$\theta$

theta

Angle of intersection

rad

$\Delta\theta_1$

Dtheta_1

Angle traveled by the first body

rad

$\Delta\theta_2$

Dtheta_2

Angle traveled by the second body

rad

$\Delta\omega_1$

Domega_1

Angular velocity difference of the first body

rad/s

$\Delta\omega_2$

Domega_2

Angular velocity difference of the second body

rad/s

$\omega_1$

omega_1

Final angular velocity of the first body

rad/s

$\omega_2$

omega_2

Final angular velocity of the second body

rad/s

$\theta_1$

theta_1

Initial angle of the first body

rad

$\theta_2$

theta_2

Initial angle of the second body

rad

$\omega_{01}$

omega_01

Initial angular velocity of the first body

rad/s

$\omega_{02}$

omega_02

Initial angular velocity of the second body

rad/s

$t_1$

t_1

Initial time of first object

s

$t_2$

t_2

Initial time of second object

s

$t$

t

Intersection time

s

$\Delta t_1$

Dt_1

Travel time of first object

s

$\Delta t_2$

Dt_2

Travel time of second object

s

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15427, 0)

Equation

>Top, >Model

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:

$\Delta\omega_1 = \omega_1 - \omega_{01}$

$\Delta\omega = \omega - \omega_0$

Conditions

Variables

$\omega$

$\omega_1$

Final angular velocity of the first body

$rad/s$

10324

$\Delta\omega$

$\Delta\omega_1$

Angular velocity difference of the first body

$rad/s$

10326

$\omega_0$

$\omega_{01}$

Initial angular velocity of the first body

$rad/s$

10322

Relation

ID:(3681, 1)

Equation

>Top, >Model

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:

$\Delta\omega_2 = \omega_2 - \omega_{02}$

$\Delta\omega = \omega - \omega_0$

Conditions

Variables

$\omega$

$\omega_2$

Final angular velocity of the second body

$rad/s$

10325

$\Delta\omega$

$\Delta\omega_2$

Angular velocity difference of the second body

$rad/s$

10327

$\omega_0$

$\omega_{02}$

Initial angular velocity of the second body

$rad/s$

10323

Relation

ID:(3681, 2)

Equation

>Top, >Model

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:

$\Delta t_1 \equiv t - t_1$

$\Delta t \equiv t - t_0$

Conditions

Variables

$t_0$

$t_1$

Initial time of first object

$s$

10252

$t$

$t$

Intersection time

$s$

10259

$\Delta t$

$\Delta t_1$

Travel time of first object

$s$

10256

Relation

ID:(4353, 1)

Equation

>Top, >Model

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:

$\Delta t_2 \equiv t - t_2$

$\Delta t \equiv t - t_0$

Conditions

Variables

$t_0$

$t_2$

Initial time of second object

$s$

10253

$t$

$t$

Intersection time

$s$

10259

$\Delta t$

$\Delta t_2$

Travel time of second object

$s$

10257

Relation

ID:(4353, 2)

Equation

>Top, >Model

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:

$\alpha_1 \equiv \displaystyle\frac{ \Delta\omega_1 }{ \Delta t_1 }$

$\bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$

Conditions

Variables

$\Delta\omega$

$\Delta\omega_1$

Angular velocity difference of the first body

$rad/s$

10326

$\bar{\alpha}$

$\alpha_1$

Angular acceleration of the first body

$rad/s^2$

10320

$\Delta t$

$\Delta t_1$

Travel time of first object

$s$

10256

Relation

Development

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)

Equation

>Top, >Model

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:

$\alpha_2 \equiv \displaystyle\frac{ \Delta\omega_2 }{ \Delta t_2 }$

$\bar{\alpha} \equiv \displaystyle\frac{ \Delta\omega }{ \Delta t }$

Conditions

Variables

$\Delta\omega$

$\Delta\omega_2$

Angular velocity difference of the second body

$rad/s$

10327

$\bar{\alpha}$

$\alpha_2$

Angular acceleration of the second body

$rad/s^2$

10321

$\Delta t$

$\Delta t_2$

Travel time of second object

$s$

10257

Relation

Development

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)

Equation

>Top, >Model

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:

$\omega_1 = \omega_{01} + \alpha_1 ( t - t_1 )$

$\omega = \omega_0 + \alpha_0 ( t - t_0 )$

Conditions

Variables

$\omega$

$\omega_1$

Final angular velocity of the first body

$rad/s$

10324

$\alpha_0$

$\alpha_1$

Angular acceleration of the first body

$rad/s^2$

10320

$\omega_0$

$\omega_{01}$

Initial angular velocity of the first body

$rad/s$

10322

$t_0$

$t_1$

Initial time of first object

$s$

10252

$t$

$t$

Intersection time

$s$

10259

Relation

Development

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)

Equation

>Top, >Model

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:

$\omega_2 = \omega_{02} + \alpha_2 ( t - t_2 )$

$\omega = \omega_0 + \alpha_0 ( t - t_0 )$

Conditions

Variables

$\omega$

$\omega_2$

Final angular velocity of the second body

$rad/s$

10325

$\alpha_0$

$\alpha_2$

Angular acceleration of the second body

$rad/s^2$

10321

$\omega_0$

$\omega_{02}$

Initial angular velocity of the second body

$rad/s$

10323

$t_0$

$t_2$

Initial time of second object

$s$

10253

$t$

$t$

Intersection time

$s$

10259

Relation

Development

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)

Equation

>Top, >Model

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:

$\theta = \theta_1 + \omega_{01} ( t - t_1 )+\displaystyle\frac{1}{2} \alpha_1 ( t - t_1 )^2$

$\theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\alpha_0$

$\alpha_1$

Angular acceleration of the first body

$rad/s^2$

10320

$\theta_0$

$\theta_1$

Initial angle of the first body

$rad$

10308

$\omega_0$

$\omega_{01}$

Initial angular velocity of the first body

$rad/s$

10322

$t_0$

$t_1$

Initial time of first object

$s$

10252

$t$

$t$

Intersection time

$s$

10259

Relation

Development

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)

Equation

>Top, >Model

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:

$\theta = \theta_2 + \omega_{02} ( t - t_2 )+\displaystyle\frac{1}{2} \alpha_2 ( t - t_2 )^2$

$\theta = \theta_0 + \omega_0 ( t - t_0 )+\displaystyle\frac{1}{2} \alpha_0 ( t - t_0 )^2$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\alpha_0$

$\alpha_2$

Angular acceleration of the second body

$rad/s^2$

10321

$\theta_0$

$\theta_2$

Initial angle of the second body

$rad$

10309

$\omega_0$

$\omega_{02}$

Initial angular velocity of the second body

$rad/s$

10323

$t_0$

$t_2$

Initial time of second object

$s$

10253

$t$

$t$

Intersection time

$s$

10259

Relation

Development

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)

Equation

>Top, >Model

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:

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:

$\theta = \theta_1 +\displaystyle\frac{ \omega_1 ^2- \omega_{01} ^2}{2 \alpha_1 }$

$\theta = \theta_0 +\displaystyle\frac{ \omega ^2- \omega_0 ^2}{2 \alpha_0 }$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\omega$

$\omega_1$

Final angular velocity of the first body

$rad/s$

10324

$\alpha_0$

$\alpha_1$

Angular acceleration of the first body

$rad/s^2$

10320

$\theta_0$

$\theta_1$

Initial angle of the first body

$rad$

10308

$\omega_0$

$\omega_{01}$

Initial angular velocity of the first body

$rad/s$

10322

Relation

Development

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)

Equation

>Top, >Model

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:

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:

$\theta = \theta_2 +\displaystyle\frac{ \omega_2 ^2- \omega_{02} ^2}{2 \alpha_2 }$

$\theta = \theta_0 +\displaystyle\frac{ \omega ^2- \omega_0 ^2}{2 \alpha_0 }$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\omega$

$\omega_2$

Final angular velocity of the second body

$rad/s$

10325

$\alpha_0$

$\alpha_2$

Angular acceleration of the second body

$rad/s^2$

10321

$\theta_0$

$\theta_2$

Initial angle of the second body

$rad$

10309

$\omega_0$

$\omega_{02}$

Initial angular velocity of the second body

$rad/s$

10323

Relation

Development

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)

Equation

>Top, >Model

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:

$\Delta\theta_1 = \theta - \theta_1$

$\Delta\theta = \theta - \theta_0$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\Delta\theta$

$\Delta\theta_1$

Angle traveled by the first body

$rad$

10310

$\theta_0$

$\theta_1$

Initial angle of the first body

$rad$

10308

Relation

ID:(3680, 1)

Equation

>Top, >Model

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:

$\Delta\theta_2 = \theta - \theta_2$

$\Delta\theta = \theta - \theta_0$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\Delta\theta$

$\Delta\theta_2$

Angle traveled by the second body

$rad$

10311

$\theta_0$

$\theta_2$

Initial angle of the second body

$rad$

10309

Relation

ID:(3680, 2)

Equation

>Top, >Model

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:

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:

$a_1 = r \alpha_1$

$a = r \alpha$

Conditions

Variables

$a$

$a_1$

First body acceleration

$m/s^2$

10264

$\alpha$

$\alpha_1$

Angular acceleration of the first body

$rad/s^2$

10320

$r$

Radio

$m$

9884

Relation

Development

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)

Equation

>Top, >Model

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:

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:

$a_2 = r \alpha_2$

$a = r \alpha$

Conditions

Variables

$a$

$a_2$

Second body acceleration

$m/s^2$

10265

$\alpha$

$\alpha_2$

Angular acceleration of the second body

$rad/s^2$

10321

$r$

Radio

$m$

9884

Relation

Development

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)