183_notes:examples:calcgravforce

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183_notes:examples:calcgravforce [2014/07/17 21:13] – pwirving183_notes:examples:calcgravforce [2018/02/09 18:33] (current) – [Solution] hallstein
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 ===== Example: Calculating the gravitational force exerted by the Earth on the Moon. ===== ===== Example: Calculating the gravitational force exerted by the Earth on the Moon. =====
  
-At a particular moment in time the Moon is located $\langle 1.9e8, 0, -1.9e8 \rangle$ m in a coordinate system in which the origin is located at the center of the Earth.+At a particular moment in time the Moon is located $\langle 1.9\times10^8, 0, -1.9\times10^8 \rangle$ m in a coordinate system in which the origin is located at the center of the Earth.
  
 Determine the gravitational force exerted by the Earth on the Moon. Determine the gravitational force exerted by the Earth on the Moon.
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 === Facts ==== === Facts ====
-  * The relative position vector from the Earth to the Moon is $\langle 1.9e8, 0, -1.9e8 \rangle$ m+  * The relative position vector from the Earth to the Moon is $\langle 1.9\times10^8, 0, -1.9\times10^8 \rangle$ m
   * Earth is origin of coordinate system ⟨0,0,0⟩ m   * Earth is origin of coordinate system ⟨0,0,0⟩ m
-  * G, the gravitational constant = $ 6.7e-11 Nm^2/kg^2 $+  * G, the gravitational constant = $ 6.7\times10^{-11} Nm^2/kg^2 $
      
  
 === Lacking === === Lacking ===
  
-  * The mass of the Earth is not given but can be [[http://lmgtfy.com/?q=mass+of+the+earth|found online]] ($5.9e24 kg$) +  * The mass of the Earth is not given but can be [[http://lmgtfy.com/?q=mass+of+the+earth|found online]] ($5.9\times10^{24} kg$) 
-  * The mass of the Moon is not given but can be [[http://lmgtfy.com/?q=mass+of+the+moon|found online]] ($7.3e22 kg$)+  * The mass of the Moon is not given but can be [[http://lmgtfy.com/?q=mass+of+the+moon|found online]] ($7.3\times10^{22} kg$)
      
 === Approximations & Assumptions === === Approximations & Assumptions ===
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 === Representations === === Representations ===
  
-  * Gravitational Force:   $ {F}_g = G\dfrac{{M}_Em}{({R}_E+y)^2}  $+  * Gravitational Force:   $ \vec{F}_{grav} = -G\dfrac{{M}_E\,M_{m}}{|\vec{r}|^2}\hat{r} $
      
-{{course_planning:earthmoon.jpg}} +{{183_notes:earth-moon-gravitation.png?400}}
    
 ==== Solution ==== ==== Solution ====
  
- First calculate the distance between moon and earth+In order to find the gravitational force we must first calculate the center to center distance between the moon and the earth
  
  |→rM−E|=√(rM−Ex)2+(rM−Ey)2+(rM−Ez)2  |→rM−E|=√(rM−Ex)2+(rM−Ey)2+(rM−Ez)2
  
- $$                = \sqrt{(1.9 x 10^8 m)^2 + (0m)^2 + (-1.9 x 10^8 m)^2}$$+ $$                = \sqrt{(1.9 \times 10^8 m)^2 + (0m)^2 + (-1.9 \times 10^8 m)^2}$$
  
  =2.7x108m  =2.7x108m
  
  
- Find the unit vector of →rM−E+We also must find the direction of this force. The direction of the force will be in the same direction as the radius vector. We can find the direction of a vector by computing the unit vector of →rM−E
  
  ˆrM−E=→rM−E|→rM−E|  ˆrM−E=→rM−E|→rM−E|
  
- $$                = \dfrac{\langle 1.9 x 10^8, 0, -1.9 x 10^8 \rangle m}{2.7 x 10^8 m}$$+ $$                = \dfrac{\langle 1.9 \times 10^8, 0, -1.9 \times 10^8 \rangle m}{2.7 \times 10^8 m}$$
  
  =⟨0.7,0,−0.7⟩  =⟨0.7,0,−0.7⟩
  
- You now have everything needed to calculate the gravitational force exerted by the Earth on the Moon:+Adding both the direction of the radius and the length of the radius to the mass of the Earth and the mass of the Moon and the gravitational constant we now have all of the variables needed to compute the gravitational force  exerted by the Earth on the Moon. The force between the Earth and the moon is the same as the gravitational force exerted by the Earth on the Moon.
  
- →FM−E=→FgravonMbyE+ →FM−E=→FgravonMbyE This is the representation we identified for gravitational force. There is a minus in front of the G as the direction of the gravitational force is opposite to the direction of the unit vector ˆr, which points from object 1 (Moon) to object 2(Earth).
  
- =−GmMmE|→rM−E|2ˆrM−E+ =−GmMmE|→rM−E|2ˆrM−E Input all the values identified for the various variables.
    
- $$                = (6.7 x 10^{-11} Nm^2/kg^2)\dfrac{(7.3 x 10^{22} kg)(5.9 x 10^{24} kg)}{(2.7 x 10^8 m)}\langle 0.7,0,-0.7 \rangle$$+ $$                = (6.7 \times 10^{-11} Nm^2/kg^2)\dfrac{(7.3 \times 10^{22} kg)(5.9 \times 10^{24} kg)}{(2.7 \times 10^8 m)^2}\langle -0.7,0,+0.7 \rangle$$  
 + 
 +Results in the magnitude of the force by unit vector (direction).
  
- $$                = 1.0 x 10^{29}N \langle 0.7,0,-0.7 \rangle$$+ $$                = 4.0 \times 10^{20}N \langle -0.7,0,+0.7 \rangle$$
  
- $$                = \langle 7.0 x 10^{28}, 0, -7.0 x 10^{28} \rangle N $$+Results in the vector force, with the x,y,z components interpretable.
  
 + =⟨−2.8×1020,0,2.8×1020⟩N
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