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| 183_notes:localg [2021/02/04 23:30] – [The Gravitational Acceleration] stumptyl | 183_notes:localg [2024/01/11 20:56] (current) – hallstein |
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| ==== The Gravitational Acceleration ==== | ==== The Gravitational Acceleration ==== |
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| Countless experiments near the surface of the Earth have shown that the force that the Earth exerts on a system with mass is the product of the system's mass ($m$) and the local gravitational acceleration ($\vec{g}$).where we have defined "up" as positive $y$-direction and the magnitude of the gravitational acceleration ($g$) is equal to **9.81 $\dfrac{m}{s}$.** | Countless experiments near the surface of the Earth have shown that the force that the Earth exerts on a system with mass is the product of the system's mass ($m$) and the local gravitational acceleration ($\vec{g}$).where we have defined "up" as positive $y$-direction and the magnitude of the gravitational acceleration ($g$) is equal to **9.81 $\dfrac{m}{s^2}$.** |
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| | {{ 183_notes:week2_m2m.png?350}} |
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| Mathematically, we represent this force like this: | Mathematically, we represent this force like this: |
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| $$\vec{F}_{Earth} = m\vec{g}$$ | $$\vec{F}_{Earth} = m\vec{g}$$ |
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| {{ 183_notes:localg.png?400}} | |
| where the local gravitational acceleration is directed towards the center of the Earth. In your typical "flat-Earth" models,((By "flat-Earth", I mean [[http://en.wikipedia.org/wiki/Geographical_distance#Flat-surface_formulae|the distance over which the Earth is curved is much larger than any distance the system will travel]] not that [[https://en.wikipedia.org/wiki/Modern_flat_Earth_societies|the Earth is truly flat as some might think]].)) you will say the gravitational acceleration points "downward", which we typically consider to be the negative $y$-direction. In this case, | where the local gravitational acceleration is directed towards the center of the Earth. In your typical "flat-Earth" models,((By "flat-Earth", I mean [[http://en.wikipedia.org/wiki/Geographical_distance#Flat-surface_formulae|the distance over which the Earth is curved is much larger than any distance the system will travel]] not that [[https://en.wikipedia.org/wiki/Modern_flat_Earth_societies|the Earth is truly flat as some might think]].)) you will say the gravitational acceleration points "downward", which we typically consider to be the negative $y$-direction. In this case, |
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| $$\vec{g} = \langle 0, -g, 0\rangle \approx \langle 0, -9.81, 0\rangle \dfrac{m}{s}$$ | $$\vec{g} = \langle 0, -g, 0\rangle \approx \langle 0, -9.81, 0\rangle \dfrac{m}{s^2}$$ |
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| We also accept some variation in $\vec{g}$ from [[http://en.wikipedia.org/wiki/Gravity_anomaly|place to place]]. | We also accept some variation in $\vec{g}$ from [[http://en.wikipedia.org/wiki/Gravity_anomaly|place to place]]. |
| ==== Motion of Systems Due to Near-Earth Gravitational Forces ==== | ==== Motion of Systems Due to Near-Earth Gravitational Forces ==== |
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| As you have read, the [[183_notes:momentum_principle|motion of a system depends on the net force]] acting on that system. If you can reasonably assume that a system interacts solely with the Earth such that the only force acting on that system is the local gravitational force, then the net force on that system is just the gravitational force. The motion of such a system is independent of the mass of the system. | As you have read, the [[183_notes:momentum_principle|motion of a system depends on the net force]] acting on that system. If you can reasonably assume that a system interacts solely with the Earth such that the only force acting on that system is the local gravitational force, then the net force on that system is just the gravitational force. ****The motion of such a system is independent of the mass of the system.**** |
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| The momentum of the system changes through the momentum principle, but the motion (how the position of the system changes) only depends on how the velocity changes. When the system only interacts with the Earth, this velocity change only depends on the gravitational acceleration. This can be summarized mathematically like this: | The momentum of the system changes through the momentum principle, but the motion (how the position of the system changes) only depends on how the velocity changes. When the system only interacts with the Earth, this velocity change only depends on the gravitational acceleration. This can be summarized mathematically like this: |
| === When are these equations useful? === | === When are these equations useful? === |
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| The previous two equations((Notice that these equations are identical to the [[183_notes:constantf|constant force motion equations]] with the gravitational force plugged in for $\vec{F}_{net}$.)) imply that the motion of objects near the surface of the Earth is independent of the mass of the object (provided you can neglect other forces). They are the basis for [[http://en.wikipedia.org/wiki/Projectile_motion | analyzing the motion of projectiles]]. But are they actually useful? | __//The previous two equations((Notice that these equations are identical to the [[183_notes:constantf|constant force motion equations]] with the gravitational force plugged in for $\vec{F}_{net}$.)) imply that the motion of objects near the surface of the Earth is independent of the mass of the object (provided you can neglect other forces)//__. They are the basis for [[http://en.wikipedia.org/wiki/Projectile_motion | analyzing the motion of projectiles]]. But are they actually useful? |
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| Galileo was the first to predict that the motion of objects near the Earth (where the Earth is the sole interaction) was independent of the mass of the object. His [[http://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pisa_experiment|supposed experiments at the Leaning Tower of Pisa]] confirmed these predictions and helped to reject the [[https://en.wikipedia.org/wiki/Aristotelian_physics|current thinking at the time, which was due to Aristotle]]. | Galileo was the first to predict that the motion of objects near the Earth (where the Earth is the sole interaction) was independent of the mass of the object. His [[http://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pisa_experiment|supposed experiments at the Leaning Tower of Pisa]] confirmed these predictions and helped to reject the [[https://en.wikipedia.org/wiki/Aristotelian_physics|current thinking at the time, which was due to Aristotle]]. |
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| {{ youtube>5C5_dOEyAfk?large }} | {{ youtube>5C5_dOEyAfk?large }} |
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| So, when you can reasonably assume that the major interaction between the system and the surroundings is the gravitational interaction with the Earth, these equations can be useful for getting a decent idea of the motion of the system. | So, when you can reasonably assume that the major interaction between the system and the surroundings is the gravitational interaction with the Earth, these equations can be useful for getting a decent idea of the motion of the system. |