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Consider a massless spring with spring constant $k$. Weights are attached to both ends with masses $m$, and $Mrespectively$. This system is placed on a horizontal surface so that weight of mass $Mlies$ on the surface and the spring with the second weight points up. The system is in&nbspequilibrium (i.e. top weight does not oscillate) and length of the spring in this state is $l$. How much do we have to compress the spring so that the weight of mass $M$ jumps up when it is released? Consider only vertical motion.

Measure the local gravitational acceleration with at least two different methods. Then compare these two methods in detail.

• Write down the equations for a throw in a homogeneous gravitational field (you don't need to prove them but you need to know how to use them). Design a machine that will throw an item and determine the angle of approach and the velocity. You can throw with the item with a spring, determine its spring constant, mass of the object and calculate the kinetic energy and thus the velocity of the item. What do you think is the precision of the your value of the velocity and angle? Put the boundaries determined by this error into the equations and show in what boundaries we can expect the distance of the landing from the origin to be.Throw the item with your device at least five times and determine the distance of the landing and what are the boundaries within which you are certain of your distance? Show if your results fit into your predictions. (For a link to video with a throw you get a bonus point!)
• Tie a pendulum with an amplitude of $x$, which effectively oscillates harmonically but the frequency of its oscillations depends on the maximum displacement $x_{0}$

$$x(t) = x_0 \cos\left[\omega(x_0) t\right]\,, \quad \omega(x_0) = 2\pi \left(1 - \frac{x_0^2}{l_0^2}\right)\,,$$

where $l_{0}is$ some length scale. We think that are letting go of the pendulum from $x_{0}=l_{0}⁄2$ but actually it is from $x_{0}=l_{0}(1+ε)⁄2$. B By how much does the argument of the cosine differ from 2π after one predicted period? How many periods will it take for the pendulum to displaced to the other side than which we expect? Tip Argument of the cosine will in that moment differ from the expected one by more than π ⁄ 2.

• Take a pen into your hand and let it stand on its tip on the table. Why does it fall? And what will determine if it will fall to the right or to the left? Why can't you predict a die throw even though the laws of physics should predict it? When you play billiard is the inability to finish the game only due to being incapable of doing all the neccessary calculations? Write down your answers and try to enumerate physics phenomenons that occur in daily life which are unpredictable even if we know the situation well.

A passanger on a ferry forgot to set the parking brake. Assume that the axis of the car is aligned with the axis of the ferry, and that because of waves the ferry is undergoing a harmonic motion, i.e. $φ(t)=Φ\sin\left(ωt)$. How far from the edge of the ferry can the passenger park the car without worrying about it falling into the sea? Assume that the maximal amplitude of oscillations is slowly increasing from zero to Φ.

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?