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Constant angular speed
Storyboard
To describe how the angle evolves over time, it's necessary to analyze its variation throughout time.
The relationship between the change in angle equals the arc angle traveled in the elapsed time, which, when divided by that time, becomes the angular velocity.
When considering a finite time interval, the angular velocity represents the average angular velocity during that interval.
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Angular velocity in graphical form
Description
The average angular velocity is defined as the angle traversed in the elapsed time. As rotation requires an axis, it is drawn orthogonal to the disk that represents the rotating body. To integrate the axis, the angular velocity is defined as a vector in which the magnitude is the angle traversed per unit of time and the direction is defined based on the direction of the axis:
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Constant angular speed
Concept
A situation that can arise is when the angular velocity is constant, which means that the angle covered increases proportionally to the elapsed time. In other words, using , this can be expressed as:
$\omega=\omega_0$
It is important to note that angular velocity is always measured relative to a reference frame. In this case, the constant angular velocity is with respect to the reference frame being used for measurement.
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Angle time for constant angular speed and initial time
Image
In the case of constant angular velocity and known initial time, the angle can be calculated using the following formula:
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )$ |
The formula is graphically represented below:
This formula is useful for calculating the angle rotated by an object in situations where both the angular velocity and initial time are known. The constancy of the angular velocity indicates that the magnitude of the angular velocity does not change with time. The initial time is the reference time from which the elapsed time is measured. Therefore, the angle rotated by the object can be calculated directly by multiplying the angular velocity by the elapsed time from the initial time.
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Tangential Speed
Description
If an object is subjected to a mode of maintaining a constant radius, it will rotate as indicated in the figure. Upon observing the figure, one would notice that the mass undergoes a translational motion with a tangential velocity that is equal to the radius times the angular velocity:
However, if the element connecting the object to the axis is cut, the object will continue to move tangentially in a straight line.
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Tangential speed, right hand rule
Image
The orientation of the tangential velocity can be obtained using the right-hand rule. If the fingers point towards the axis of rotation and then are curled towards the position vector (radius), the thumb will point in the direction of the tangential velocity:
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Model
Concept
Variables
Parameters
Selected parameter
Calculations
Equation
$ \vec{v} = \vec{\omega} \times \vec{r} $
&v = &omega x &r
$ \Delta s=r \Delta\theta $
Ds = r * Dtheta
$ \Delta s \equiv s - s_0 $
Ds = s - s_0
$ \Delta t \equiv t - t_0 $
Dt = t - t_0
$ \Delta\theta = \theta - \theta_0 $
Dtheta = theta - theta_0
$ \omega \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$
omega = Dtheta / Dt
$ \theta = \theta_0 + \omega_0 ( t - t_0 )$
theta = theta_0 + omega_0 * ( t - t_0 )
$ v = r \omega $
v = r * omega
$ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$
v_m = Ds / Dt
$ \bar{v} = v_0$
v_m = v_0
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Elapsed time
Equation
To describe the motion of an object, we need to calculate the elapsed time. This magnitude is obtained by measuring the initial time and the final time of the motion. The duration is determined by subtracting the initial time from the final time.
This is mathematically represented as
 $ \Delta t \equiv t - t_0 $ |
where $\Delta t$ is the duration, $t$ is the final time, and $t_0$ is the initial time.
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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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Mean angular speed
Equation
To estimate the displacement of an object, it's necessary to know its the angular Speed ($omega$) as a function of the time ($t$). Therefore, the the mean angular velocity ($omega_m$) is introduced, defined as the ratio between the angle variation ($\Delta\theta$) and the time elapsed ($\Delta t$).
To measure this, a system like the one shown in the image can be used:
To determine the average angular velocity, a reflective element is placed on the axis or on a disk with several reflective elements, and the passage is recorded to estimate the length of the arc $\Delta s$ and the angle associated with the radius $r$. Then the time difference when the mark passes in front of the sensor is recorded as $\Delta t$. The average angular velocity is determined by dividing the angle traveled by the time elapsed.
The equation that describes the average angular velocity is:
| $ \omega \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$ |
The definition of the mean angular velocity ($omega_m$) is considered as the angle variation ($\Delta\theta$),
| $ \Delta\theta = \theta - \theta_0 $ |
and the time elapsed ($\Delta t$),
| $ \Delta t \equiv t - t_0 $ |
The relationship between both is defined as the mean angular velocity ($omega_m$):
| $ \omega \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$ |
It should be noted that the average velocity is an estimation of the actual angular velocity. The main problem is that:
If the angular velocity varies during the elapsed time, the value of the average angular velocity can be very different from the average angular velocity.
Therefore, the key is:
Determine the velocity in a sufficiently short elapsed time to minimize its variation.
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Angle for constant angular velocity
Equation
When the angular velocity is constant (represented as $\omega_0$), the average angular velocity (represented as $omega$) is equal to the instantaneous angular velocity, so that:
$\omega = \omega_0$
In this case, the angle covered by an object can be calculated as a function of time by recalling that it is associated with the difference between the current and initial angles, and the current and initial times.
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )$ |
If the angular velocity is constant, equal to $\omega_0$, the average angular velocity will be equal to:
$\omega = \omega_0$
Therefore, with the angle traveled:
| $ \Delta\theta = \theta - \theta_0 $ |
and the elapsed time:
| $ \Delta t \equiv t - t_0 $ |
we have that the equation for the average angular velocity:
| $ \omega \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$ |
can be written as:
$\omega_0 = \omega = \displaystyle\frac{\Delta\theta}{\Delta t} = \displaystyle\frac{\theta - \theta_0}{t - t_0}$
so that by rearranging, we obtain:
| $ \theta = \theta_0 + \omega_0 ( t - t_0 )$ |
This equation represents a straight line in the angle-time space.
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Distance traveled
Equation
We can calculate the distance traveled in a time ($\Delta s$) from the starting position ($s_0$) and the position ($s$) using the following equation:
| $ \Delta s \equiv s - s_0 $ |
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Average Speed
Equation
The mean Speed ($\bar{v}$) can be calculated from the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$) using:
| $ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
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Average and constant speed
Equation
When the velocity is constant, then trivially the average velocity is equal to that constant velocity. In other words, the constant velocity ($v_0$) is equal to the mean Speed ($\bar{v}$):
| $ \bar{v} = v_0$ |
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Arc traveled
Equation
The position the distance traveled in a time ($\Delta s$) in a circular motion can be calculated from the angle variation ($\Delta\theta$) and the radius ($r$) of the orbit using the following formula:
| $ \Delta s=r \Delta\theta $ |
If an object is at a distance equal to the radius of an axis and performs a rotation at an angle
| $ \Delta\theta = \theta - \theta_0 $ |
,
it will have traveled a length
| $ \Delta s \equiv s - s_0 $ |
.
This arc can be calculated by multiplying the radius by the angle, which is
| $ \Delta s=r \Delta\theta $ |
.
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Speed and angular speed
Equation
If we divide the arc length equation
| $ \Delta s=r \Delta\theta $ |
by the elapsed time, we obtain the equation that allows us to calculate the velocity along the orbit, which is called the tangential velocity:
| $ v = r \omega $ |
Since the translational velocity is equal to
| $ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
and the path expressed as the arc of a circle is
| $ \Delta s=r \Delta\theta $ |
and using the definition of the average angular velocity
| $ \omega \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$ |
we have that
$v=\displaystyle\frac{\Delta s}{\Delta t}=r\displaystyle\frac{\Delta\theta}{\Delta t}=r\omega$
Since the relation is general, it can be applied for instantaneous values, resulting in:
| $ v = r \omega $ |
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Tangential speed, vector shape
Equation
Angular velocity is defined as a vector whose direction coincides with the axis of rotation. Since the radius of rotation and angular velocity are perpendicular to the tangential velocity, it can be expressed as the cross product of the angular velocity and the radius of rotation:
| $ v = r \omega $ |
This can be written concisely as:
| $ \vec{v} = \vec{\omega} \times \vec{r} $ |
Since the tangential velocity is
| $ v = r \omega $ |
we can calculate the tangential vector using the cross product with the axis versor, denoted as $hat{n}$, and the radial versor, denoted as $hat{r}$:
$\hat{t} = \hat{n} \times \hat{r}$
Therefore, if we define
$\vec{v}=v\hat{t}$
,
$\vec{r}=r\hat{r}$
and
$\vec{\omega}=\omega\hat{n}$
,
then we can express the velocity as
$\vec{v}=v\hat{t}=v\hat{n}\times\hat{r}=r\omega\hat{n}\times\hat{r}=\vec{\omega}\times\vec{r}$
which means that
| $ \vec{v} = \vec{\omega} \times \vec{r} $ |
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