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.

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    <h3> Statement </h3>
    <p>
        $3.1.17.$ 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.
    </p>

    <h3>Solution</h3>
    <p>

$$N=mg\cos\varphi$$
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:
$$\fbox{$A=R\sqrt{\frac{2\Delta}{3N+\Delta}}$}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$A=R\sqrt{\frac{2\Delta}{3N+\Delta}}$$
    </p>
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	  Dzikan Mikita<br>
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