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183_notes:localg [2015/09/03 13:55] – [Motion of Systems Due to Near-Earth Gravitational Forces] caballero | 183_notes:localg [2021/02/04 23:30] – [The Gravitational Acceleration] stumptyl |
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| Section 2.5 in Matter and Interactions (4th edition) |
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===== Constant Force: Gravitational Force near Earth ===== | ===== Constant Force: Gravitational Force near Earth ===== |
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You've read that the [[183_notes:momentum_principle|net force acting on an systems will change the system's momentum]], but until now you haven't considered any particular forces. The first force that you will consider is the one that results from the interaction between objects with mass: [[http://en.wikipedia.org/wiki/Gravity|the gravitational force]]. | You've read that the [[183_notes:momentum_principle|net force acting on an systems will change the system's momentum]], but until now you haven't considered any particular forces. The first force that you will consider is the one that results from the interaction between objects with mass: [[http://en.wikipedia.org/wiki/Gravity|the gravitational force]]. |
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For now, you will consider only the motion of systems near the surface of the Earth. Near the surface of the Earth, we observe that the gravitational force is a constant vector.((This is mostly true. There are small variations due to changes in the density of the Earth's crust in different regions. These [[http://en.wikipedia.org/wiki/Gravity_anomaly|gravitational anamolies]] were mapped by the [[http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment|GRACE experiment]].)) Later, you will find that the gravitational force near the surface of the Earth is an approximation to the more general description of [[183_notes:gravitation|the gravitational force between objects]]. | __//For now, you will consider only the motion of systems near the surface of the Earth. Near the surface of the Earth, we observe that the gravitational force is a constant vector.//__((This is mostly true. There are small variations due to changes in the density of the Earth's crust in different regions. These [[http://en.wikipedia.org/wiki/Gravity_anomaly|gravitational anamolies]] were mapped by the [[http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment|GRACE experiment]].)) Later, you will find that the gravitational force near the surface of the Earth is an approximation to the more general description of [[183_notes:gravitation|the gravitational force between objects]]. |
==== Lecture Video ==== | ==== Lecture Video ==== |
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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 ($\vec{g}$). Mathematically, we represent this force like this: | 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 $\dfrac{m}{s}$.** |
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| Mathematically, we represent this force like this: |
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$$\vec{F}_{Earth} = m\vec{g}$$ | $$\vec{F}_{Earth} = m\vec{g}$$ |
$$\vec{g} = \langle 0, -g, 0\rangle \approx \langle 0, -9.81, 0\rangle \dfrac{m}{s}$$ | $$\vec{g} = \langle 0, -g, 0\rangle \approx \langle 0, -9.81, 0\rangle \dfrac{m}{s}$$ |
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where we have defined "up" as positive $y$-direction and the magnitude of the gravitational acceleration ($g$) is equal to 9.81 $\dfrac{m}{s}$. 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]]. |
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The figure on the right represents a typical [[https://en.wikipedia.org/wiki/Free_body_diagram|force body diagram]] for two systems falling near the surface of the Earth (where we have neglected any interactions due to the air). Notice that while the two systems experience different forces, they experience the [[183_notes:acceleration|same acceleration]]. | The figure on the right represents a typical [[https://en.wikipedia.org/wiki/Free_body_diagram|force body diagram]] for two systems falling near the surface of the Earth (where we have neglected any interactions due to the air). Notice that while the two systems experience different forces, they experience the [[183_notes:acceleration|same acceleration]]. |
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$$\vec{r}_f = \vec{r}_i + \vec{v}_i \Delta t + \dfrac{1}{2}\vec{g}\Delta t^2$$ | $$\vec{r}_f = \vec{r}_i + \vec{v}_i \Delta t + \dfrac{1}{2}\vec{g}\Delta t^2$$ |
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| When we choose the +$y$-direction to be up from the surface of the Earth, the gravitational acceleration is given by $\vec{g} = \langle 0,-9.8,0\rangle\dfrac{m}{s}$. This leads you to a set of 4 kinematic equations, which you might be familiar from other courses, that describe the motion of a system that can be reasonably assumed to experience just the gravitational interaction. |
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| $$v_{f,x} = v_{i,x}$$ |
| $$v_{f,y} = v_{i,y} -g\Delta t$$ |
| $${x}_f = {x}_i + {v}_{i,x} \Delta t$$ |
| $${y}_f = {y}_i + {v}_{i,y} \Delta t - \dfrac{1}{2}g\Delta t^2$$ |
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| === 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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{{ 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. |
===== Examples ===== | ===== Examples ===== |
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* [[183_notes:examples:Finding the time of flight of a projectile]] | * [[183_notes:examples:Finding the time of flight of a projectile]] |
* [[183_notes:examples:Finding the range of projectile]] | * [[183_notes:examples:Finding the range of projectile]] |