$3.1.17.$ The normal pressure force of a small body changes from $N$ to $N + \Delta$, $\Delta \ll N$ when it oscillates in a well of radius $R$ near its equilibrium position. Determine the amplitude of oscillations of this body.
$$ N=mg\cos\varphi $$ Let's use the approximation, for $\varphi\ll1$, $\cos\varphi=1-\frac{\varphi^2}{2}$: $$ N\approx mg\left(1-\frac{\varphi^2}{2}\right)=mg\left(1-\frac{A^2}{2R^2}\right)\quad(1) $$ $$ N+\Delta=m(a+g)=m\left(\frac{\upsilon^2}{R}+g\right) $$ Law of conservation of energy $$ mgR(1-\cos\varphi)\approx mgR\frac{\varphi^2}{2}=\frac{m\upsilon^2}{2} $$ $$ N+\Delta=m\left(\frac{gR\varphi^2}{R}+g\right)=mg(1+\varphi^2) $$ $$ N+\Delta=mg\left(1+\frac{A^2}{R^2}\right)\quad(2) $$ Next we solve the system of equations $(1)$ and $(2)$ and obtain that: $$ \fbox{$A=R\sqrt{\frac{2\Delta}{3N+\Delta}}$} $$
$$A=R\sqrt{\frac{2\Delta}{3N+\Delta}}$$