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| 183_notes:grav_accel [2014/09/10 14:38] – caballero | 183_notes:grav_accel [2021/02/05 00:02] (current) – [The Gravitational Force and the Momentum Principle] stumptyl |
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| | Section 3.2 and 3.3 in Matter and Interactions (4th edition) |
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| ===== Gravitational Acceleration ===== | ===== Gravitational Acceleration ===== |
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| $$\vec{a}_2 = -G\dfrac{m_1}{|\vec{r}|^2}\hat{r}$$ | $$\vec{a}_2 = -G\dfrac{m_1}{|\vec{r}|^2}\hat{r}$$ |
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| The resulting expression is the acceleration that object 2 experiences due to it's gravitational interaction with object 1. Notice that the acceleration of object 2 depends only on the mass of object 1 ($m_1$), and relative position of object 2 with respect to object 1 ($\vec{r}$). It also points towards object 1, which indicates that the object 2 is attracted (and will thus experience an acceleration along the line between object 1 and 2). | //__The resulting expression is the acceleration that object 2 experiences due to it's gravitational interaction with object 1__//. Notice that the acceleration of object 2 depends only on the mass of object 1 ($m_1$), and relative position of object 2 with respect to object 1 ($\vec{r}$). It also points towards object 1, which indicates that the object 2 is attracted (and will thus experience an acceleration along the line between object 1 and 2). |
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| So, in general: | So, __in general__: |
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| $$\vec{a} = -G\dfrac{m}{|\vec{r}|^2}\hat{r}$$ | $$\vec{a} = -G\dfrac{m}{|\vec{r}|^2}\hat{r}$$ |
| ==== The Local Gravitational Acceleration revisited ==== | ==== The Local Gravitational Acceleration revisited ==== |
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| Earlier you read that the [[183_notes:localg|local gravitational acceleration]] was given by $\vec{g} \approx \langle 0,-9.81,0\rangle m/s$ or, rather that the magnitude of the acceleration was $g \approx -9.81 m/s.$ It turns out this value can be predicted by Newton's model for gravitational interactions, which demonstrates that the force that keeps us grounded on Earth is the very same force that hold [[http://en.wikipedia.org/wiki/Solar_System|planets in orbit]] and is responsible for the [[http://en.wikipedia.org/wiki/Star_formation|formation of stars]]. | Earlier you read that the [[183_notes:localg|local gravitational acceleration]] was given by $\vec{g} \approx \langle 0,-9.81,0\rangle m/s$ or, rather that the magnitude of the acceleration was $g \approx 9.81 m/s.$ It turns out this value can be predicted by Newton's model for gravitational interactions, which demonstrates that the force that keeps us grounded on Earth is the very same force that hold [[http://en.wikipedia.org/wiki/Solar_System|planets in orbit]] and is responsible for the [[http://en.wikipedia.org/wiki/Star_formation|formation of stars]]. |
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| For simplicity, let's take the downward vertical direction to be positive. Let's compute the acceleration due gravity at the surface of the Earth. Here the [[http://lmgtfy.com/?q=mass+of+the+earth|mass of the Earth]] is roughly $5.97\times10^{24} kg$ and [[http://lmgtfy.com/?q=radius+of+the+earth|the radius of the Earth]] is $6.38\times10^6 m$. | For simplicity, let's take the downward vertical direction to be positive. Let's compute the acceleration due gravity at the surface of the Earth. Here the [[http://lmgtfy.com/?q=mass+of+the+earth|mass of the Earth]] is roughly $5.97\times10^{24} kg$ and [[http://lmgtfy.com/?q=radius+of+the+earth|the radius of the Earth]] is $6.38\times10^6 m$. |
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| $$a_y=G\dfrac{M_{Earth}}{R_{Earth}} = \left(6.67384 \times 10^{-11} \dfrac{m^3}{kg\:s^2}\right)\left(\dfrac{5.97\times10^{24}\:kg}{(6.38\times10^6\:m)^2}\right) = 9.80 \dfrac{m}{s^2}$$ | $$a_y=G\dfrac{M_{Earth}}{R^2_{Earth}} = \left(6.67384 \times 10^{-11} \dfrac{m^3}{kg\:s^2}\right)\left(\dfrac{5.97\times10^{24}\:kg}{(6.38\times10^6\:m)^2}\right) = 9.80 \dfrac{m}{s^2}$$ |
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| which is pretty close to the value we often use. In fact, the gravitational acceleration fluctuates a few percent over the surface of the Earth due to [[http://en.wikipedia.org/wiki/Gravity_anomaly|gravitaitonal anomalies]]. The variations in the Earth's crust that are primarily responsible for these anomalies were mapped by the [[http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment|GRACE Experiment]]. | which is pretty close to the value we often use. In fact, the gravitational acceleration fluctuates a few percent over the surface of the Earth due to [[http://en.wikipedia.org/wiki/Gravity_anomaly|gravitaitonal anomalies]]. The variations in the Earth's crust that are primarily responsible for these anomalies were mapped by the [[http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment|GRACE Experiment]]. |
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