1... urban walk
3 points
Matěj walks across the street with constant velocity. Every 7 minutes a tram going in opposite direction passes, while every 10 minutes a tram going in his direction passes. We assume that trams ride in both directions with the same frequency. What is the frequency?
2... warm reachability into the ball
3 points
Imagine you have subcooled homogeneous metal ball that you have just taken out of a freezer set to very low temperature. You want to find out how fast the ball temperature will increase if you put it in a warm room. It would be a university-level problem. Because of that, we made it easier for you. We ask about how deep into the ball will the „warm area“ reach. You can estimate it using dimensional analysis. We know relevant parameters of the ball - its density
3... border
6 points
Imagine an aquarium in the shape of a cube with edge length
Bonus: Find frequency of small oscillations of the partition. Assume, that mass of the partition is
4... splash
8 points
Consider a free water droplet with radius of
5... bouncing ball
9 points
We spin a rigid ball in the air with angular velocity
P... 1 second problems
9 points
Suggest several ways to slow down the Earth so that we would not have to add the leap second to certain years. How much would it cost?
E... thirty centimeters tone
12 points
Everyone has ever tried out of boredom to strum on a long ruler sticking out of the edge of a school-desk. Choose the right model of frequency versus the part of the length of the ruler which is sticking out and prove it experimentally. Also, describe other properties of the ruler.
Note: Allow vibration only for outsticking part of the ruler by fixing its position above the table.
Instructions for Experimental TasksS... heavenly-mechanic
10 points
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- Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius
. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically. - Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass
in a spherically symmetrical central-force field. \[ \] Where is the length of the radius vector, is the total energy, is angular momentum, and is the potential energy of the mass point. - Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.