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Inclined plane
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When a body is placed on an inclined plane, it begins to slide under the action of gravity. However, its vertical velocity component is smaller than in free fall because part of the acceleration is projected onto the direction parallel to the plane, reducing its velocity in the vertical axis.
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Potential Energy
Storyboard
If a body is moved by defeating a force on a given path, energy can be stored that can then accelerate the body by imparting a speed and thereby kinetic energy. Stored energy has the potential to accelerate the body and is therefore called potential energy.
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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
When an object moves from a height $h_1$ to a height $h_2$, it covers the difference in height
$h = h_2 - h_1$
thus, the potential energy
becomes equal to
Examples
When a body is placed on an inclined plane and there is no friction preventing it from sliding, it begins to accelerate under the action of gravity. However, the gravitational force acting vertically decomposes into a component parallel to the plane, whose magnitude is:
$F_p = m_g g \sin\theta$
This depends on the gravitational mass ($m_g$), the gravitational Acceleration ($g$), and the angle of the inclined plane ($\phi$). This force gives rise to the potential energy:
expressed as a function of the path traveled on the inclined plane ($s$).
The total Energy ($E$) corresponds to the sum of the total Kinetic Energy ($K$) and the potential Energy ($V$):
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.
In the case of an inclined plane, the path traveled on the inclined plane ($s$) is proportional to the height gained as a function of the angle of the inclined plane ($\phi$). Therefore, the potential Energy ($V$) is expressed as a function of the path traveled on the inclined plane ($s$), the angle of the inclined plane ($\phi$), the mass ($M$), and the gravitational Acceleration ($g$):
The total Energy ($E$) of a the inertial Mass ($m_i$) moving at the speed ($v$) on an inclined plane, under the effect of gravity generated by its the mass ($M$) with the gravitational Acceleration ($g$), on a plane with the angle of the inclined plane ($\phi$) and covering a path the path traveled on the inclined plane ($s$), is expressed as:
The masses that Newton used in his principles are related to the inertia of bodies, which leads to the concept of the inertial Mass ($m_i$).
Newton's law, which is linked to the force between bodies due to their masses, is related to gravity, hence known as the gravitational mass ($m_g$).
Empirically, it has been concluded that both masses are equivalent, and therefore, we define
Einstein was the one who questioned this equality and, from that doubt, understood why both 'appear' equal in his theory of gravity. In his argument, Einstein explained that masses deform space, and this deformation of space causes a change in the behavior of bodies. Thus, masses turn out to be equivalent. The revolutionary concept of space curvature implies that even light, which lacks mass, is affected by celestial bodies, contradicting Newton's theory of gravitation. This was experimentally demonstrated by studying the behavior of light during a solar eclipse. In this situation, light beams are deflected due to the presence of the sun, allowing stars behind it to be observed.
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