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## gravitational field

### (3 points)6. Series 37. Year - 1. ballons with Martin

A car is standing on a straight road, with a freely floating helium balloon tied inside. Suddenly, the car starts to accelerate with acceleration $a=5{,}0 \mathrm{km\cdot min^{-2}}$. By what angle will the balloon be deflected from the vertical line? What is the direction of the deflection?

Martin would like to hang on a balloon behind a car.

### (10 points)1. Series 37. Year - P. rocket

Using current technology, how much fuel would it take to carry an object of mass $m=1 \mathrm{kg}$ into low Earth orbit?

The leprechaun wanted to save on rocket fuel.

### (9 points)3. Series 36. Year - P. absurd pendulum

What phenomena can affect the measurement of gravitational acceleration using a pendulum? Estimate how many valid digits your result would have to contain to measure them. Consider also the phenomena that you usually neglect.

Kačka was wondering what she could write in the discussion.

### (10 points)2. Series 36. Year - P. planetary atmosphere

What parameters does a planet need to have to keep its atmosphere comparable to the Earth? What conditions are essential for the planet to gain such an atmosphere?

### (13 points)5. Series 35. Year - E. it's already going

Measure the moment of inertia of a cylinder (regarding its main axis) and a ball (with respect to the axis passing through its center) by rolling them on an inclined plane.

Karel imagined participants rolling.

### (3 points)3. Series 35. Year - 1. Where my center of gravity is?

We can find an unofficial interpretation that the red, blue and white colors on the Czech flag symbolize blood, sky (i.e. air) and purity. Find the position of the center of mass of the flag interpreted in this way, assuming that purity is massless. The aspect ratio is $3:2$ and the point where all three parts meet is located exactly in the middle. Look up the blood and air densities.

Bonus: Try to calculate the position of the center of mass of the Slovak flag as accurately as possible. You can use different approximations.

Matěj likes to have fun with flags.

### (9 points)3. Series 35. Year - P. artificial gravitation

How could artificial gravity be implemented on a spaceship? What would be the advantages and disadvantages depending on the different characteristics of the spacecraft? Is it realistic to have gravity in different directions on different floors of the spaceship or for it to change rapidly, as we can sometimes see in sci-fi movies when „artificial gravity fails“?

Karel was day-dreaming while watching sci-fi.

### (8 points)1. Series 35. Year - 5. mechanically (un)stable capacitor

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

Vašek wanted to grill you on a capacitor.

### (3 points)5. Series 34. Year - 1. the charge of the Earth

Find the total electric charge, that the Earth would need to let all electrons close to its surface fly away. How would this charge differ if it had to deflect protons?

Karel likes planetary problems.

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