A long non-stretchable rope is thrown over a stationary block. At one end of the rope hangs a weight of mass $m = 10 kg$, the other end is grasped by a monkey of mass $M = 30 kg$. With what acceleration relative to the rope must the monkey move to remain at the same height above the Earth's surface? Neglect the mass of the rope and the friction in the block.

A weightless rope is thrown over a log. A body of mass $m$ is tied to one end of the rope, and a force $F$ directed vertically downward is applied to the other end. Because of the friction between the rope and the log, the minimum value of the force at which the body $m$ hangs motionless is $\alpha mg$, where $\alpha$ is less than unity and independent of the unit $m$. Find the minimum value of the force $F$ at which body $m$ will rise upward.

A weightless rope is thrown over a log. A body of mass $m$ is tied to one end of the rope. Some force $F$ is applied vertically downward to the other end. Because of the friction between the rope and the log, the minimum value of the force that lifts the body up is $\alpha mg$, where $\alpha$ is a coefficient greater than one and independent of $m$. Find the minimum value of force $F$ sufficient to hold body $m$ stationary.

On the edge of a cart of mass $M$ and length $l$ standing on a smooth table there is a small cube of mass $m$. The cube is pushed with a constant force $F$ directed horizontally to the opposite edge of the cart. In what time will the cube reach the opposite edge of the cart? The coefficient of friction between the cube and the cart is $\mu$. The cube does not roll over.

A boat is pulled along the water with a constant force $\bar{F}$. Its velocity is constant and equal to $\bar{v}$. What is the distance $S$ the boat will travel in the water after the force ceases, if the force resisting the boat's motion is proportional to its velocity relative to the water?

A small ball of mass m suspended on an inextensible thread in the field of gravity rotates in the vertical plane. At the upper point of the trajectory, the tension of the thread is zero. Find the tension of the thread at the lower point of the trajectory.

A thin hoop of mass m slides frictionlessly on a smooth horizontal surface with velocity $v_{0}$, without rotation. The hoop axis is horizontal, vector $v_{0}$ lies in the plane of the hoop. At time $t=0$, the smooth surface is replaced by a rough surface with friction coefficient $\mu$. Find the steady-state velocity of the center of the hoop.

A car passes a curve of a road, which has the form of a circle (the angle of rotation is $90^{ \circ}$), with a constant speed $v$ and in time $t$. What is the average force acting on the car during the turn if the mass of the car is $m$?

A ball is suspended on an inextensible thread of length $l$ in the field of gravity with free fall acceleration $g$. At what speed $v$ must the suspension point be moved horizontally for the ball to make a complete revolution in the vertical plane?

A car is driving on a bridge having the shape of a parabola. The bridge height $h = 5m$, horizontal length $l = 60m$. Find the ratio of the force of pressure of the car on the road at the top of the bridge to its weight on a level road, if the car is traveling on the bridge at a constant speed $v = 54 km/h$.

A light spring with stiffness coefficient $k$ is attached at one end to a vertical axis around which it can rotate freely, and at the other end to a small weight of mass $m$. The whole system is on a horizontal smooth table, the spring is not stretched. A shock imparts to the weight a velocity $v_{0}$ directed perpendicular to the spring. Find the minimum and maximum distance from the weight to the axis if the velocity of the weight at the maximum distance from the axis is $v_{1}$.

A horizontal disk of radius $R = 10 m$ rotates about its axis with constant angular velocity $\omega = 0.2 rad/s$. A motorcyclist rides along the edge of the disk with constant velocity $v = 36 km/h$ relative to the disk in the direction opposite to the direction of rotation of the disk. What should be the coefficient of friction between the tires of the motorcycle and the surface of the disk, so that such a movement is possible?

At what maximum speed $v_{max}$ can a motorcyclist drive around a curve in the road with radius $R = 120 m$ if the coefficient of friction between the motorcycle tires and the ground $\mu = 0.25$?

How do the Sun's attraction and the Earth's rotation affect the readings of a spring scale that measures the weight of a body at the equator at noon and midnight? Consider that Earth's axis of rotation is perpendicular to the plane of the orbit.

Why are almost all satellites launched from west to east? Estimate the velocities that must be communicated to a satellite to be launched into a low circular orbit along the equator, west to east and east to west.