2.28. Under the action of gravity $\dot{g}$, directed opposite to the $O Z$ axis, a body of unit mass slides down a spiral:

$$
x=a \cos \varphi, y=a \sin \varphi, z=b(2 \pi-\varphi)
$$

from point $A(a, 0,2 \pi b)$ to point $B(a, 0,0)$.

Find the work of the field during such a displacement.
Solution. The vector field $\vec{F}$ of gravitational force has components:

$$
\vec{F}=(0,0,-g)
$$

where $g \approx 9.81 \frac{\mathrm{M}}{c^{2}}$.

Obviously, $\operatorname{rot} \vec{F}=\overrightarrow{0}$, hence this field is potential. Let's determine its potential:

$$
\varphi=\int_{0}^{x} 0 d t+\int_{0}^{y} 0 d t-\int_{0}^{z} g d t+C=C-g \cdot z \quad(C=\text { const })
$$

The work of the field $\vec{F}$ when moving a body from $A$ to $B$ is equal to the difference in potentials at these points:

$$
A=\varphi(B)-\varphi(A)=C-(C-g \cdot 2 \pi b)=2 \pi g b
$$