An aircraft is flying at \(800 \mathrm{~km} \mathrm{~h}^{-1}\) in latitude \(55^{\circ} \mathrm{N}\). Find the angle through which it must tilt its wings to compensate for the horizontal component of the Coriolis force.

First, we need to convert the given speed to meters per second (m/s) to make the calculation easier. $$ 800 \mathrm{~km} \mathrm{~h}^{-1} = 800 \cdot 1000 \mathrm{~m} \mathrm{~h}^{-1} \cdot \frac{1}{3600} \mathrm{~h} \mathrm{~s}^{-1} = 222.22 \mathrm{~m} \mathrm{~s}^{-1} $$

The Earth's angular velocity (\(\omega\)) can be calculated using the following formula: $$ \omega = \frac{2\pi}{T} $$ Where \(T\) is the Earth's period, which is approximately 24 hours. We need to convert \(T\) to seconds: $$ T = 24 \cdot 60 \cdot 60 = 86400 \mathrm{~s} $$ Now, we can find the Earth's angular velocity: $$ \omega = \frac{2\pi}{86400} \approx 7.27 \cdot 10^{-5} \mathrm{~rad} \mathrm{~s}^{-1} $$

The formula for Coriolis acceleration is given by: $$ a_c = 2v\omega\sin\phi $$ Where \(v\) is the aircraft's speed in m/s, \(\omega\) is the Earth's angular velocity in rad/s, and \(\phi\) is the latitude in radians. Let's convert \(55^\circ\) to radians: $$ \phi = 55^\circ \cdot \frac{\pi}{180} \approx 0.9599 \mathrm{~rad} $$ Now we can calculate the Coriolis acceleration: $$ a_c = 2 \cdot 222.22 \cdot 7.27 \cdot 10^{-5} \cdot \sin (0.9599) \approx 0.0325 \mathrm{~m} \mathrm{~s}^{-2} $$

The lift force of the wings should be equal to the horizontal component of the Coriolis acceleration acting on the aircraft's mass. We can write down the equation: $$ F_L = ma_c $$ Where \(F_L\) is the lift force, \(m\) is the aircraft's mass, and \(a_c\) is the Coriolis acceleration. To find the angle of wing tilt (\(\theta\)), we will use the following relationship: $$ \sin\theta = \frac{F_L}{mg} $$ Where \(g\) is the acceleration due to gravity (\(9.81 \mathrm{~m} \mathrm{~s}^{-2}\)). By substituting \(F_L = ma_c\), we can rewrite the equation as: $$ \sin\theta = \frac{a_c}{g} $$ So, we can find the angle \(\theta\) as follows: $$ \theta = \arcsin\left(\frac{a_c}{g}\right) = \arcsin\left(\frac{0.0325}{9.81}\right) $$ $$ \theta \approx 0.0033 \mathrm{~rad} $$ Now, let's convert this angle to degrees: $$ \theta = 0.0033 \cdot \frac{180}{\pi} \approx 0.19^{\circ} $$

The aircraft must tilt its wings by approximately \(0.19^{\circ}\) to compensate for the horizontal component of the Coriolis force.