mechanisms

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.

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.

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In order to carry out this measurement precisely, a stroboscopic lamp can be used, which emits flashes of light at defined intervals. By capturing a photograph at a specific instant, it is possible to determine the angular distance that the object travels during that time period. By calculating angular velocities at two consecutive moments, the change in angular velocity can be obtained, and by dividing this change by the time interval between the photographs, the average angular acceleration is derived.

The equation that describes this average angular acceleration is as follows:

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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.

In the case of constant angular acceleration, the angular velocity follows a linear relationship with respect to time:

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which is depicted in the following graph:

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With the constant Acceleration ($a_0$), the function of the angular Speed ($\omega$) describes a straight line whose slope corresponds to the angular acceleration. Together with the initial Angular Speed ($\omega_0$), the time ($t$) and the start Time ($t_0$), this relationship is expressed by the equation:

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Therefore, the area under the curve representing the total angular displacement consists of a rectangle and a triangle:

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The rectangle has a height equal to the initial angular velocity and a base corresponding to the elapsed time. The triangle has a height equal to the product of the angular acceleration and the elapsed time, with a base that is also equal to the time.

With this information, the total displacement the angle ($\theta$) can be calculated using the initial Angle ($\theta_0$), as shown below:

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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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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:

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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:

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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:

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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:

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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:

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This equation represents a straight line in the angular velocity versus time plane.

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:

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This expression corresponds to the general form of a parabola.

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:

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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:

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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:

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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:

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