The surface of the web woven by the spider is horizontal. When the spider is in its center, the web deflects, the value of deflection $x_{0}=1 mm$. A fly, once caught in the web, starts beating. Can we determine from the frequency of resonance oscillations whether the mass of the fly exceeds the mass of the spider or not?

As you know, when calculating the period of oscillation of a mathematical pendulum, you substitute $\sin \alpha$ for $\alpha$, where $\alpha$ is the angle of deviation of the pendulum string from the vertical. Find out whether the period calculated in this way is smaller or larger than the real period. A mathematical pendulum is a point mass $m$ suspended on a weightless, inextensible thread of length $l$.

A small ball of mass m, attached to a spring with stiffness coefficient $k$, makes harmonic oscillations in the direction of the spring axis with amplitude $A$. When the spring was compressed, a second spring pendulum of the same mass is placed in the path of the ball, both springs having a common axis and the equilibrium positions of the balls coinciding. Describe the further motion of the system in the following cases:
(a) The shock is central and perfectly elastic;
b) the impact is central and absolutely inelastic (i.e. the balls "stick together").
At the initial moment the second ball rests in the equilibrium position.

In a deep vertical shaft underground there are two weights of equal mass connected by a weightless non-stretchable thread, which is thrown over a fixed block. Find the period of vertical oscillation of the weights if the radius of the Earth is 6400 km, the acceleration of free fall at the surface is $10 m/s^{2}$.