Two identical trains are moving on rails along the equator on the earth in opposite directions with the same speed. They will exert the same pressure on the rails.

Step 1: Define the parameters of the system.

Let the two identical trains each have mass $m$, and let them move with speed $v$ relative to the Earth's surface along the equator in opposite directions.

Step 2: Determine the inertial speeds of the trains.

The Earth rotates about its axis with angular speed $\Omega$. In an inertial reference frame, the speed of the Earth's surface at the equator is $v_E = \Omega R$, where $R$ is the radius of the Earth.

The inertial speed of the train moving in the direction of the Earth's rotation is:

$$v_{\text{with}} = \Omega R + v$$

The inertial speed of the train moving against the direction of the Earth's rotation is:

$$v_{\text{against}} = \Omega R - v$$

Step 3: Set up the equation of motion in the radial direction.

In the inertial frame, the net inward radial force provides the necessary centripetal acceleration for circular motion of radius $R$.

$$m g_0 - N = \frac{m v_I^2}{R}$$

Here, $g_0 = \frac{GM}{R^2}$ is the acceleration due to gravity, $N$ is the normal force exerted by the rails on the train, and $v_I$ is the inertial speed of the train.

Step 4: Calculate the normal force for each train.

Solve for the normal force $N$, which is equal in magnitude to the force (and thus pressure) the train exerts on the rails.

For the train moving with the rotation:

$$N_{\text{with}} = m\left(g_0 - \frac{(\Omega R + v)^2}{R}\right)$$

For the train moving against the rotation:

$$N_{\text{against}} = m\left(g_0 - \frac{(\Omega R - v)^2}{R}\right)$$

Step 5: Compare the normal forces.

Find the difference between the two normal forces.

$$N_{\text{against}} - N_{\text{with}} = m\left(g_0 - \frac{(\Omega R - v)^2}{R}\right) - m\left(g_0 - \frac{(\Omega R + v)^2}{R}\right)$$ $$N_{\text{against}} - N_{\text{with}} = \frac{m}{R} \left[ (\Omega R + v)^2 - (\Omega R - v)^2 \right]$$ $$N_{\text{against}} - N_{\text{with}} = 4 m \Omega v$$

Since $4 m \Omega v > 0$, the train moving against the Earth's rotation exerts a greater force on the rails than the train moving with the rotation.

Answer: The statement is FALSE. The train moving against the Earth's rotation exerts a greater pressure on the rails than the train moving with the rotation, with a difference in normal force of $4 m \Omega v$.