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Translational kinetic energy
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The kinetic energy of translation is a function of the velocity achieved through the application of a force over a given time while traveling a given path.
Thus, the kinetic energy of translation is proportional to the mass of the object and the square of the velocity.
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Translational kinetic energy
Storyboard
The kinetic energy of translation is a function of the velocity achieved through the application of a force over a given time while traveling a given path. Thus, the kinetic energy of translation is proportional to the mass of the object and the square of the velocity.
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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 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 definition of the mean Acceleration ($\bar{a}$) is considered as the relationship between the speed Diference ($\Delta v$) and the time elapsed ($\Delta t$). That is,
and
The relationship between both is defined as the centrifuge Acceleration ($a_c$)
within this time interval.
Since the moment ($p$) is defined with the inertial Mass ($m_i$) and the speed ($v$),
If the inertial Mass ($m_i$) is equal to the initial mass ($m_0$), then we can derive the momentum with respect to time and obtain the force with constant mass ($F$):
$F=\displaystyle\frac{d}{dt}p=m_i\displaystyle\frac{d}{dt}v=m_ia$
Therefore, we conclude that
Examples
The work variance ($\Delta W$) is defined as the product of the force with constant mass ($F$) and the distance traveled in a time ($\Delta s$):
La variaci n del trabajo en el tiempo se denomina la potencia. Por lo general es una limitante ya que indica la velocidad que es un sistema capaz de crear/absorber energ a.
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:
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 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 proportion in which the variation of velocity over time is defined as the mean Acceleration ($\bar{a}$). To measure it, it is necessary to observe the speed Diference ($\Delta v$) and the time elapsed ($\Delta t$).
One common method for measuring average acceleration involves using a stroboscopic lamp that illuminates the object at defined intervals. By taking a photograph, one can determine the distance traveled by the object in that time. By calculating two consecutive velocities, one can determine their variation and, with the time elapsed between the photos, the average acceleration.
The equation that describes average acceleration is as follows:
It is important to note that average acceleration is an estimation of actual acceleration.
The main problem is that if acceleration varies during the elapsed time, the value of the average acceleration may differ greatly from the mean acceleration
.
Therefore, the key is to
Determine acceleration over a sufficiently short period of time to minimize variation.
In the case where the inertial Mass ($m_i$) equals the initial mass ($m_0$),
the derivative of momentum will be equal to the mass multiplied by the derivative of the speed ($v$). Since the derivative of velocity is the instant acceleration ($a$), we have that the force with constant mass ($F$) is
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:
Acceleration corresponds to the change in velocity per unit of time.
Therefore, it is necessary to define the speed Diference ($\Delta v$) in terms of the speed ($v$) and the initial Speed ($v_0$) as follows:
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