Determine the maximum velocity of a ball of a mathematical pendulum of length l moving in one plane if the amplitude of the displacement at small oscillations of the pendulum is equal to x_0.

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Solutions of Savchenko Problems in Physics
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    <h3> Statement </h3>
    <p>
        $3.1.6.$ Determine the maximum velocity of a ball of a mathematical pendulum of length $l$ moving in one plane if the amplitude of the displacement at small oscillations of the pendulum is equal to $x_0$.
    </p>

    <h3>Solution</h3>
    <p>
        The motion of a mathematical pendulum is described by the solution of the equation of harmonic oscillations

$$x = x_0\sin\omega t$$
Where $\omega$ is the cyclic frequency of oscillations
$$\omega = \sqrt{\frac{g}{l}}$$
Let's take the derivative of $x(t)$:
$$v = \dot{x}(t) = \omega \cdot x_0\cos\omega t$$
Making the maximum speed equal to
$$\boxed{\upsilon_\text{max}=x_0\omega=x_0\sqrt{\frac{g}{l}}}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$\upsilon_\text{max}=x_0\sqrt{\frac{g}{l}}$$
    </p>
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