Worksheet 14.2: Tides

Note: Once again, Latex is only working where it wants to work, for no conceivable reason and I am very open to suggestions on how to fix it.

1. Draw a circle representing the Earth (mass M‘), with 8 equally-spaced point masses, m, placed around the circumference. Also draw the Moon with mass MK to the side of the Earth. In the following, do each item pictorially, with vectors showing the relative strengths of various forces at each point. Don’t worry about the exact geometry, trig and algebra. I just want you to think about and draw force vectors qualitatively, at least initially.

1. (a)  What is the gravitational force due to the Moon, F⃗K,cen, on a point at the center of the Earth? Recall that vectors have both a magnitude (arrow length) and a direction (arrow head).
   
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3. (b)  What is the force vector on each point mass, F⃗K, due to the Moon? Draw these vectors at each point.
   
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   (c)  What is the force difference, ∆F⃗, between each point and Earth’s center? This is the tidal force.
   
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   (d)  What will this do to the ocean located at each point?
   
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7. (e)  How many tides are experienced each day at a given location located along the Moon’s orbital plane?
   
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   In a 24 hour period, there would be 2 high tides and 2 low tides.

   
   
9. (f)  Okay, now we will use some math. For the two points located at the nearest and farthest points from the Moon, which are separated by a distance ∆r compared to the Earth-Moon distance r, show that the force difference is given by \[ \Delta F = \frac{2GmM_{\circ} \Delta r}{r^3} \] Using \[ \lim_{x\to\ 0} \frac{f(x + \Delta x ) - f(x) }{\Delta x} = \frac{d}{dx} f(x) \approx \frac{ \Delta f(x) }{\Delta x} \]  \[ F_g (r) = \frac{GmM_{\circ}}{r^2} \]  \[ F (r + \Delta r) = \frac{GmM_{\circ}}{(r + \Delta r)^2} \] Because we know  \[ \lim_{x\to\ 0} \frac{f(x + \Delta x ) - f(x) }{\Delta x} = \frac{d}{dx} f(x) \approx \frac{ \Delta f(x) }{\Delta x} \] We can take the derivative of F to find: \[ \frac{d}{dx} F = -\frac{2GmM_{\circ} \Delta r}{r^3} \]

7. (g)  Compare the magnitude of the tidal force ∆FK caused by the Moon to ∆F@ caused by the Sun. Which is stronger and by how much? What happens when the Moon and the Sun are on the same side of the Earth?
   
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9. The only pertinent differences in the equation are M and r so we can compare:\[ \Delta F_k = \frac{M }{r^3} \] \[ \frac{M_{\odot} }{r^3} vs = \frac{M_{\circ} }{r^3} \] \[ \frac{2 \times 10^{33} }{(1.5 \times 10^{13})^3} vs \frac{7.3 \times 10^{25}}{(3.8 \times 10^{10})^3} \] \[5.9 \times 10^{-7} \: vs \: 1.3 \times 10^{-6} \] So the moon would have a stronger pull. If they are on the same side, both forces would pull together and add for a stronger tidal force. 
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11. (h)  How does the magnitude of ∆F caused by the Moon compare to the tidal force caused by Jupiter during its closest approach to the Earth (r « 4 AU)? \[ \frac{M_{\circ} }{r^3} vs \frac{M_{jup}}{r^3} \] \[ \frac{7.3 \times 10^{25}}{(3.8 \times 10^{10})^3} vs  \frac{1.9 \times 10^{30} }{(4 \times 1.5 \times 10^{13})^3} \] \[ 1.3 \times 10^{-6} \: vs \: 8.8 \times 10^{-12} \] So the moon would once again have a greater tidal force. 

I worked with Sean, Barra, and April on this problem.