- #1
Mk
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Problem statement
The first thing that I notice, is that
[tex]T = 2\pi\sqrt{\frac{R ^3}{g}}[/tex]
looks suspiciously like
[tex]T = 2\pi\sqrt{\frac{\ell}{g}}[/tex]
And [tex]g_x=G\frac{M}{R^3}x[/tex] looks suspiciously like:
[tex]F = G \frac{m_1 m_2}{r^2}[/tex]
[tex]F=m_1 a_1[/tex]
[tex]a_1= \frac {F}{m_1}[/tex]
[tex]a_1 = G \frac{m_2}{r^2}[/tex]
And other than that, I'm stuck. I don't even understand what this problem is asking for sure yet. And just because something "looks" like something doesn't mean necessarily it's right, but both of the equations do look quite suspicious! I would be pleased with any hints.
Very fast trains can travel from one city to another in straight subterranean tunnels. Assume that the density of the Earth is constant so that the acceleration of gravity as a function of the radial distance r from the center of the Earth is g=(GM/R^3)r, where G, M, and R are constants.
1. Show that the component of gravity along the track of the train is gx=-(GM/R^3)x where x is measured from the midpoint of the track.
2. Neglecting friction, show that the motion of the train along the track is simple harmonic motion with a period independent of the length of the track.
[tex]T_0 = 2\pi\sqrt{\frac{R ^3}{g}}[/tex]
The first thing that I notice, is that
[tex]T = 2\pi\sqrt{\frac{R ^3}{g}}[/tex]
looks suspiciously like
[tex]T = 2\pi\sqrt{\frac{\ell}{g}}[/tex]
And [tex]g_x=G\frac{M}{R^3}x[/tex] looks suspiciously like:
[tex]F = G \frac{m_1 m_2}{r^2}[/tex]
[tex]F=m_1 a_1[/tex]
[tex]a_1= \frac {F}{m_1}[/tex]
[tex]a_1 = G \frac{m_2}{r^2}[/tex]
And other than that, I'm stuck. I don't even understand what this problem is asking for sure yet. And just because something "looks" like something doesn't mean necessarily it's right, but both of the equations do look quite suspicious! I would be pleased with any hints.