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A hair with a lot of curls can be approximated by a spring (in other words it can be described by the exact same parameters – radius, inclination and material constants). How much longer is the hair if it hangs vertically in a gravitational field compared to the case that it lies horizontally on a table? Assume that the inclination is very small.

• Strings.

Using dimensional analysis determine the dependance of the frequency of oscillatons of a string if you know that it depends on its length $l$, on the tension $F$ in the string and on its linear density $ρ_{l}$.

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• Downward.

You have a dumbell which consists of a short rod and two heavy discs. You wrap a string around the rod and let the dumbell fall while holding the string. What is the velocity of the dumbell? The discs have mass $M$ and radius $R$. The radius of the rod is $r$ and you can neglect its mass.

Two straws are connected so that the resulting object is of the shape of letter V. This object is then supported on its sides so that it can swing. Determine the stability conditions (the object should not slide) and also the period of oscillations of this system. The radius of the straws is $r$.

Lukáš found an old sofa spring of force constant $k$, coil radius $r$, length $l$ and the number of coils $n$. Since he was bored, he connected the spring to electric current $I$. How did the action change the force constant of the spring?

What is the optimum speed of walking person to put a bridge into a maximum amplitude? Define parameters needed and then solve.

You can buy a roller on which surface a small bumps are located. The bumps then hit an edge of steel wafer, which have different length. In a song recorded on the roller you can find all tones in some interval (say, C major scale). Can you find out the shape of the wafers?

Marek made himself before Christmas a new toy: onto a wooden circle he put two nails (both nails and centre of circle in one line) and on two springs of stiffness $k$ he placed a weight of mass $m$. The weight can slide without friction on the surface of the wooden circle. Marek placed the circle onto a table and spin it around its axis with angular velocity $ω$, while the weight was slightly moved off the centre (equilibrium). Describe the movement of the weight, and as a bonus you can calculate it.

Two weights of masses $m$ and $M$ are connected by a spring of stiffness $k$ and lies on flat smooth surface (friction can be neglected). Body of mass $m$ gets velocity $\textbf{v}$ (see picture). What will be the smallest distance between the bodies? And when it will be reached?

Kathy is on a swing (a plank suspended on 2 ropes). At high displacements she kneels down, in lowest point she gets up. This movements she periodically repeats. Ration of distance of centre of gravity from the rotation axis at kneeling down and at standing is 2^{1 ⁄ 12} ≈ 1,06. How many times Kathy must swing to double the amplitude of swinging?

The food on a transatlantic ship is prepared by a chef Thomas. He has very userful spring-based device to hold plates. The holder holds the top plate always in the same height regardless the number of plates in holder. The distance between plates is 1 cm. The weather is stormy and the plates are oscillating. What is the frequency of the oscillations?