$1.3.29.$ To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius $R$. The roadbed forms an angle $\alpha$ with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity $v$?
Find the value of acceleration $ a $:
$$ a = \sqrt{a_{\tau}^2 + a_{n}^2} $$
Let $ a_{\tau} = 0 $, then $ a_{n} = \frac{v^2}{R} $, where $ v $ is the horizontal component of the velocity directed perpendicular to the axis of rotation.
Then the required acceleration will be equal to:
$$ a = a_n $$
$$\fbox{$ a = \frac{v^2}{R} \cdot \cos^2 \alpha $}$$
$$a = \frac{v^2}{R} \cdot \cos^2 \alpha $$