Setting up a differential equation

[danielI]

So I need help with this diff eq.
A train is placed in the tunnel AB and is will roll through it under its own weight, stop and return. Show that the time for a complete round trip is the same for all tunnels and calculate it.

Using the fact that the force of gravity (inside the earth) is proportional to the distance from the center
$$F_g = m\frac{d^2x}{dt^2} = -kx$$

I also know that $$\frac{4000}{\sin\gamma} = \frac{x}{\sin\alpha}$$ and $$\frac{F}{\sin\beta} = \frac{F_g}{\sin\alpha}$$

Is this enough? I can't delete $$\gamma$$ and $$\beta$$ which I think I should be able to.

Cheers

(the red dot is the position of the train at an arbitrary time)

 

[HallsofIvy]

Is your "$\gamma$" the angle that looks like a "v" in the picture? If so then obviously $\alpha+ \beta+ \gamma= \pi$. You can eliminate either $\beta$ or $\gamma$ but not both. You don't want to- one of them will be your variable.

 

[tacman]

To set up a differential equation for this problem, we can use Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration. In this case, the only force acting on the train is the force of gravity, which is proportional to the distance from the center of the earth. Therefore, we can write the following differential equation:

m\frac{d^2x}{dt^2} = -kx

where m is the mass of the train, x is the position of the train, t is time, and k is a constant representing the proportionality between the force of gravity and the distance from the center of the earth.

To solve this differential equation, we can use the initial conditions given in the problem. The train is initially at point A, so we can set x(0) = x_A, where x_A is the distance from the center of the earth to point A. The train will then roll through the tunnel and come to a stop at point B, so we can set x(t_1) = x_B, where t_1 is the time it takes for the train to reach point B.

Next, the train will start moving in the opposite direction due to the force of gravity, and eventually come to a stop at point A again. We can set x(t_2) = x_A, where t_2 is the time it takes for the train to return to point A.

Now, we can solve the differential equation using these initial conditions and find the time it takes for a complete round trip, which is t_1 + t_2. This time will be the same for all tunnels, as it is only dependent on the initial position of the train and the proportionality constant k.

To calculate the time, we can use the given equations \frac{4000}{\sin\gamma} = \frac{x}{\sin\alpha} and \frac{F}{\sin\beta} = \frac{F_g}{\sin\alpha}. These equations relate the angles and distances involved in the problem. By substituting these values into our differential equation and solving for t_1 + t_2, we can calculate the time for a complete round trip.

In conclusion, the differential equation m\frac{d^2x}{dt^2} = -kx, along with the given initial conditions and equations, can be used to calculate the