$3.2.3.$ 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?
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 $$ \boxed{\frac{T_1}{T_2}=2} $$
The period will be halved