A closed vessel is filled with water. The area of the lower base of the vessel $S_{1}=100 m^{2}$, the area of the upper base $S_{2}=200 \: m^{2}$ the height of the vessel $h=50 m$. The water pressure force on the upper base $F_{2}=100 \: N$. find the pressure force on the lower base of the vessel if the density of water $\rho_{0} =1000 \: kg/m^{3}$, the acceleration of free fall $g=10 \: m/s^{2}$

A small ball thrown with initial velocity $v_{0}$ at an angle a to the horizon hit a vertical wall moving towards it with horizontally directed velocity $v(t)$ and bounced back to the point from which it was thrown. Determine at what time t after the throw did the ball collide collision of the ball with the wall? Neglect friction losses.

A tube of diameter $d$ is lowered into a vessel with water. A ball of the same diameter is placed in the tube. The center of the ball is at a depth $h$. Find the density of the substance of the ball. There is no gap between the pipe and the ball. The frictional force between them is zero.

Two identical bathyspheres float in suspension, the first at depth $2d$, the second at depth $d$ ($d = 1 km$). At the initial moment of time the first bathysphere discharges ballast and starts to float. When it floats to depth d, the second bathysphere bathysphere discharges ballast and begins to float. The first bathysphere appeared on the water surface at $T = 10 s$ earlier than the second one. It is known that the first bathysphere moved with almost constant velocity $v_{0} = 1 m/s$ for the second half of its path. Find the ejective force. The mass of the bathysphere without ballast is $m$. (The force of resistance to the motion of the bathysphere from the water side can be considered directly proportional to the speed of the bathysphere.

Lead and aluminum balls of equal radius $r$ are connected by a weightless non-stretchable thread, the length of which is much greater than the size of the balls. The balls were lowered in a vessel with glycerol. They then came to motion with zero initial velocity. The drag force on the balls is proportional to their velocity, and the coefficient of proportionality is the same for both balls. Find the tension force of the thread at the steady-state velocity of the balls. The densities of aluminum and lead are $\rho_{1}$ and $\rho_{2}$.

How to lower from the roof of height $h = 16 m$ a weight of mass $m = 45 kg$ with the help of a rope, whose breaking resistance $T$ is $400 N$? The velocity of the body at the moment of impact with the ground must not exceed $v_{max} = 7 m/s$. The length of the rope is slightly greater than the height of the house.

A snake lying on the floor starts to rise with vertical velocity $v$. Find the pressure force on the floor if the snake is homogeneous, its mass is $m$, length $l$

A ball falls in the air from a very high height and hits the ground absolutely elastically. Find the acceleration of the ball immediately after the impact, considering the drag force acting on the ball from the air to be proportional to its velocity.

To remove a cork stuck in the neck of a thermos, you poke a thin awl into the neck of the thermos. At what angle $\theta$ to the axis of the thermos can the awl be poked in without fear that the cork will fall through the thermos? The coefficient of friction of the cork against the walls of the neck is $\mu = 0.5$.

A body of mass $m$ is pressed against the ceiling with force $F$ directed at an angle $\alpha$ to the horizon. At what values of the coefficient of friction $\mu$ between the body and the ceiling will the body remain stationary?

It is known that the force of dry friction between two bodies is practically independent of the area of contact between these bodies. However, the deeper the cork is inserted into the neck of a bottle, the more difficult it is to remove it. Why?

A train approaching a station with speed $v = 60 km/h$ starts to slow down. What is the minimum time to stop the train so that the suitcases lying on the shelves do not move? Consider that during braking the train moves with constant acceleration, and the coefficient of friction $\mu$ of the suitcase against the shelf is 0.2.

Two coupled cars of mass $m = 10 t$ each roll along the rails. The coefficient of sliding friction between the wheels of the cars and the rails is $\mu = 0.3$, rolling friction can be neglected. Suddenly, the axles of the rear wagon are jammed so that its wheels stop rotating. Find the acceleration of the cars and the force acting on the coupling (the device connecting the cars) after the axles jam. How much time will elapse before the cars stop if their initial velocity is $v = 10 m/s$?

Two bars with masses $m_{1}$ and $m_{2}$ are lying on a horizontal surface connected by a weightless non-stretchable thread. An external force F is applied horizontally. Determine the tension force of the thread if the coefficients of sliding friction between the surface and the bars are $\mu_{1}$ and $\mu_{2}$, respectively.

A force $F$ acts on the right end of an inextensible rope of length $l$ located on a perfectly smooth horizontal table in the horizontal direction along the rope. Find the tension of the rope at a distance $x$ from its left end.