A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?

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Solutions of Savchenko Problems in Physics
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    <h3> Statement </h3>
    <p>
        $3.2.3.$ A weight is oscillating vertically on a rubber cord. How many times will the period of vertical oscillations of the weight change if it is suspended on the same cord folded in half?
    </p>

    <h3>Solution</h3>
    <p>
        We find the period of oscillations on a standard cord as the period of oscillations of a spring pendulum

$$T_1=2\pi\sqrt{\frac{m}{k}}$$
When we fold the cord, we change its length and width and, accordingly, its stiffness $k'$ changes
$$T_2=2\pi\sqrt{\frac{m}{k'}}$$
Since the spring is divided into two and the length of each half is $l/2$ and the stiffness of each of them is $k_1=2k$; by adding the two halves in parallel we obtain a parallel connection
$$k'=2k_1=4k$$
We substitute the oscillation period of the “new” cord into the expression for $T_2$
$$T_2=2\pi\sqrt{\frac{m}{4k}}=\pi\sqrt{\frac{m}{k}}$$
From where we obtain the ratio of the periods of oscillations
$$\boxed{\frac{T_1}{T_2}=2}$$

    </p>
  <p style="text-align: right; font-style: italic; font-size: 14;">
	  Dzikan Mikita<br>
	  Aliaksandr Kanashenka
    </p>

    <h4>Answer</h4>
    <p>
        The period will be halved
    </p>


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