Some planet consists entirely of an incompressible liquid of density $\rho$. The temperature in the depth of the planet is constant. Find the dependence of pressure on depth. The radius of the planet $R$.

What energy must be given to a spacecraft of mass 1 t in order for it to enter a circular orbit with radius $R = 16000 km$ after starting from the Earth's surface? Consider that the potential energy of a body in the Earth's gravity field at a distance from the Earth's center $R \geq R_{earth}$ is $E_{n} = - \frac{GmM_{earth}}{R}$, where $M_{z}$ is the Earth's mass.

Calculate the first space velocity at the launch from the surface of Jupiter, if it is known that one of its satellites rotates in a nearly circular orbit of radius $r = 10^{6} km$ with period $T = 7.1 days$. The radius of Jupiter is $R = 7 \cdot 10^{4} km$.

In the absence of air resistance, a man on the surface of the Earth can push a nucleus to $l = 22 m$. Can he push the same nucleus on the surface of an asteroid so that it becomes a satellite? The mass of the asteroid is $M = 3 \cdot 10^{13} t$, radius $R = 10 km$.

Let there is a ball with radius equal to the Earth radius and mass equal to the Earth mass. Some small body is located on the ball. The ball begins to rotate around an axis perpendicular to the radius connecting the body to the center of the ball. Describe what will happen to the body as the angular velocity of the ball slowly increases. Between
surfaces of the body and the ball there is friction.

Determine the radius $r$ of the orbit of an Earth satellite that is above the same point on the Earth's surface at all times.

A particle of mass $m$ moves under the action of some force along a circular orbit of radius $R$ with constant velocity. The potential energy $U$ of the particle in the field of this force depends only on the distance to the center of the orbit as follows:
a) $U(R) = kR (k > 0)$;
b) $U(R)=kR^{2} \: (k>0)$.
Find the velocity of the particle in cases a) and b).

Compare the potential energies of a body on the surface of the Earth and on the surface of the Moon, Accept that the acceleration of free fall on the Moon is six times less than on the Earth, and the radius of the Moon is three times less than the radius of the Earth.

A satellite flies around the earth in a circular orbit with radius $R = 25600 km$. How many times is the satellite's speed different from the first space velocity?

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?

A beaker of mass $m = 100 g$ and volume $V = 200 cm^{3}$ is suspended upside down from a dynamometer, with the edges of the beaker touching the surface of water and the inside volume of the beaker filled with water. Why does the water not pour out of the beaker? What does the dynamometer show?

A hermetically closed absolutely rigid vessel of height $H$ is filled to the top with water. The water pressure at the upper wall of the vessel is zero. At the bottom of the vessel there is a small air bubble. How will the pressure of the water in the vessel change if the bubble floats up? Assume that water is incompressible. Neglect the solubility of air in water.

The king ordered Archimedes to measure as much gold as an elephant weighs. Such large scales were nowhere to be found. Archimedes had a raft at his disposal. How did Archimedes solve the problem?

A body completely immersed in a liquid floats at some depth in equilibrium. What physical parameters determine the stability of this equilibrium? Write down the condition of stability of equilibrium, considering that the temperature of the liquid does not depend on the depth.

Two metal balls (a lead ball and an iron ball) are balanced on different weights. When the balls are lowered into water, the equilibrium is not disturbed. Try to find an explanation for this phenomenon.