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Energy Units
Definition 
Las unidades de la energía se han nombrado en honor a James Joule que descubrió la equivalencia entre energía térmica y mecánica. La unidad es igual a\\n\\n
$J=\displaystyle\frac{kg,m^2}{s^2}$
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The energy required for an object to change its angular velocity from $\omega_1$ to $\omega_2$ can be calculated using the definition
Applying Newton's second law, this expression can be rewritten as
$\Delta W=I \alpha \Delta\theta=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta$
Using the definition of angular velocity
we get
$\Delta W=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta=I \omega \Delta\omega$
The difference in angular velocities is
$\Delta\omega=\omega_2-\omega_1$
On the other hand, angular velocity itself can be approximated with the average angular velocity
$\omega=\displaystyle\frac{\omega_1+\omega_2}{2}$
Using both expressions, we obtain the equation
$\Delta W=I \omega \Delta \omega=I(\omega_2-\omega_1)\displaystyle\frac{(\omega_1+\omega_2)}{2}=\displaystyle\frac{I}{2}(\omega_2^2-\omega_1^2)$
Thus, the change in energy is given by
$\Delta W=\displaystyle\frac{I}{2}\omega_2^2-\displaystyle\frac{I}{2}\omega_1^2$
This allows us to define kinetic energy as
The work variance ($\Delta W$) required for an object to change from the initial Angular Speed ($\omega_0$) to the angular Speed ($\omega$) is obtained by applying a the torque ($T$) that produces an angular displacement the difference of Angles ($\Delta\theta$), according to:
Applying Newton's second law for rotation, in terms of the moment of inertia for axis that does not pass through the CM ($I$) and the mean Angular Acceleration ($\bar{\alpha}$):
this expression can be rewritten as:
$\Delta W = I \alpha \Delta\theta$
or, using the difference in Angular Speeds ($\Delta\omega$) and the time elapsed ($\Delta t$):
we get:
$\Delta W = I\displaystyle\frac{\Delta\omega}{\Delta t} \Delta\theta$
Using the definition of the mean angular velocity ($\bar{\omega}$) and the time elapsed ($\Delta t$):
results in:
$\Delta W = I\displaystyle\frac{\Delta\omega}{\Delta t} \Delta\theta = I\omega \Delta\omega$
where the difference in Angular Speeds ($\Delta\omega$) is expressed as:
On the other hand, the angular velocity can be approximated by the average angular velocity:
$\bar{\omega}=\displaystyle\frac{\omega_1 + \oemga_2}{2}$
By combining both expressions, we obtain the equation:
$\Delta W = I \omega \Delta\omega = I(\omega_2 - \omega_1) \displaystyle\frac{(\omega_1 + \omega_2)}{2} = \displaystyle\frac{I}{2}(\omega_2^2 - \omega_1^2)$
Therefore, the change in energy is expressed as:
$\Delta W = \displaystyle\frac{I}{2}\omega_2^2 - \displaystyle\frac{I}{2}\omega_1^2$
This allows us to define the rotational kinetic energy as:
Expanding the concept to a longer path involves summing up the energy required for each path element:
$\bar{W}=\displaystyle\sum_i \vec{F}_i\cdot\Delta\vec{s}_i$
However, the value of this equation represents just an average energy required or generated. The precise energy is obtained when the steps become very small, allowing the force to be considered constant within them:
$W=\displaystyle\sum_i \mbox{lim}_{\Delta\vec{s}_i\rightarrow\vec{0}}\vec{F}_i\cdot\Delta\vec{s}_i$
In this limit, the energy corresponds to the integral along the traveled path, giving us:
Examples
The concept of energy was initially introduced in thermodynamics with the purpose of quantifying the amount of heat that could be converted into mechanical work. In a representative experiment, friction was generated by sliding a surface against a cable subjected to a force. This cable traveled a the distance traveled in a time ($\Delta s$) which, when multiplied by the applied force the force with constant mass ($F$), resulted in the generated work the work variance ($\Delta W$):
Since both the force with constant mass ($F$) and the distance traveled in a time ($\Delta s$) are actually vectors, this expression can be generalized using the scalar product between the force ($\vec{F}$) and the path traveled (vector) ($\Delta\vec{s}$), yielding the work fraction ($\Delta W$):
In other words, only the component of the force that acts in the direction of the displacement effectively contributes to the energy transfer.
Las unidades de la energ a se han nombrado en honor a James Joule que descubri la equivalencia entre energ a t rmica y mec nica. La unidad es igual a\\n\\n
$J=\displaystyle\frac{kg,m^2}{s^2}$
Carnot was the first to describe energy in terms of the path and the necessary force to traverse it. Progressing along a path with a force requires or generates energy. This corresponds to the equation:
In the continuous limit, the sum can be represented as an integral:
The translational Kinetic Energy ($K_t$) is determined based on the speed ($v$) and the inertial Mass ($m_i$), according to:
5288 is associated with 6290 and not with 8762, even though they are numerically equal. The energy that an object possesses is a direct consequence of the inertia that had to be overcome to set it in motion.
The kinetic energy of rotation ($K_r$) is a function of the angular Speed ($\omega$) and of a measure of inertia represented by the moment of inertia for axis that does not pass through the CM ($I$):
The total Kinetic Energy ($K$) can have translational and/or rotational components. Therefore, it is expressed as the sum of the translational Kinetic Energy ($K_t$) and the kinetic energy of rotation ($K_r$):
In a more complex system, the total kinetic energy is equal to the sum of the kinetic energies of the individual parts
In a more complex system, the total potential energy is equal to the sum of the potential energies of the individual parts
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