Savchenko Solutions

Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#3.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $3.1.5.$ The length of the string of a mathematical pendulum is $l$, the mass of the ball is $m$. Determine the force acting on the ball as it deviates from the equilibrium position by $x$ in the case where $x \ll l$. How does the potential energy of the ball depend on $x$?

    <h3>Solution</h3>
    <p>
        When the pendulum swings away, it is acted upon by a restoring force $F$ acting in the direction opposite to the direction of the velocity, which is restoring and equal to

$$OX:F=-mg\sin\varphi=\fbox{$-mg\frac{x}{l}$}$$
Considering that the oscillations are harmonic, their amplitude is not large, and the magnitude of the deviation is much less than the length of the thread: $(x\ll l)$. Let us use the expression for small angles $\cos\alpha\approx1-\frac{\alpha^2}{2};\quad\sin\alpha\approx\alpha$
$$1-\cos\alpha\approx\frac{x^2}{2l^2}$$
Changing the position of the ball vertically
$$\Delta h = l \cdot (1-\cos\alpha)=\frac{x^2}{2l}$$
Where does the change in potential energy come from?
$$\Delta U = mg\cdot\Delta h = \boxed{mg\frac{x^2}{2l}}$$

Aliaksandr Melnichenka

    <h4>Answer</h4>
    <p>
        $$F=-mg\frac{x}{l};\quad U=\frac{mgx^2}{2l}$$
    </p>


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