183_notes:grav_accel

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183_notes:grav_accel [2014/09/10 14:40] – [The Local Gravitational Acceleration revisited] caballero183_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)
 +
 ===== Gravitational Acceleration ===== ===== Gravitational Acceleration =====
  
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 $$\vec{a}_2 = -G\dfrac{m_1}{|\vec{r}|^2}\hat{r}$$ $$\vec{a}_2 = -G\dfrac{m_1}{|\vec{r}|^2}\hat{r}$$
  
-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).
  
-So, in general:+So, __in general__:
  
 $$\vec{a} = -G\dfrac{m}{|\vec{r}|^2}\hat{r}$$ $$\vec{a} = -G\dfrac{m}{|\vec{r}|^2}\hat{r}$$
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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 $6.38\times10^6 m$.
  
-$$a_y=G\dfrac{M_{Earth}}{R_{Earth}} = \left(6.67384 \times 10^{-11} \dfrac{m^3}{kg\:s^2}\right)\left(\dfrac{5.97\times10^{24}\:kg}{(6.38\times10^6\:m)^2}\right) = 9.80 \dfrac{m}{s^2}$$+$$a_y=G\dfrac{M_{Earth}}{R^2_{Earth}} = \left(6.67384 \times 10^{-11} \dfrac{m^3}{kg\:s^2}\right)\left(\dfrac{5.97\times10^{24}\:kg}{(6.38\times10^6\:m)^2}\right) = 9.80 \dfrac{m}{s^2}$$
  
 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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  • Last modified: 2014/09/10 14:40
  • by caballero