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The author: Alexander P. Kuznetsov, Doctor of Sciences
1. Estimate pressure producing by a man standing on the Earth surface.
2. Estimate a value of a buoyancy force acting onto a brick in water.
3. What distance you pass making a million of steps?
4. How much water in all oceans weighs?
5. How much time is required for a modern train to pass by you?
6. Estimate a distance up to the visible horizon.
7. How many pig-pong balls could be placed in your classroom?
8. Estimate a size of asteroids, starting from which they may be regarded as spheroid-like. Assume that solidity of rocks there is of the same order as in the Earth. Is your result in agreement with the data of observation of asteroids and satellites from space probes?
1. On a board of mass M a little brick of mass m is placed. The friction coefficient between the board and the brick equals k1, and that between the board and the surface is k2. A horizontally directed force F effects the brick. Find all possible qualitatively distinct situations of the system behavior. Establish parameter intervals where these situations take place. Depict domains on the parameter plane k1, k2 corresponding to different types of dynamical behavior.
2. On a horizontal surface having the friction coefficient k a body of mass m is placed. A force F is supplied to the body, directed at some angle in respect to horizon. Enumerate all possible qualitatively distinct types of behavior of the system. Depict the respective domains in the plane of dimensionless parameters (k, mg/F). How will the character of motion be changed under effect of force of fixed absolute value, when the angle of its direction is gradually increased?
3. In a heat-isolated vessel filled with water of mass M at temperature T an experimentalist has placed a piece of ice of mass m at temperature -t. What are qualitatively distinct states of the system possible after a reach of thermal equilibrium? Depict domains on a plane of parameters (m, M) corresponding to each special state. Indicate points on the plane associated with zero final temperature.
4. A thin uniform rod of length l and density is hinged by one end in such way that the hinge is at a distance h from a surface of liquid of density . What are qualitatively distinct situations of disposition of the road? Depict domains on the parameter plane (h/l, ) associated with these situations. Consider two cases, when the hinge is placed upside or under the surface of liquid.
5. A ball of mass m is attached to a weightless rubber thread having the spring coefficient k, and its length without deformation equals l. The ball is rotated in such way that the thread is at a finite angle in respect to the vertical. Select appropriate dimensionless parameters for this problem and depict a domain, where such a kind of behavior takes place. What happens outside this domain?
6. A known experiment demonstrating inertial properties of bodies consists in the following. We pull a thread attached at the bottom point of a massive ball hanged on another thread. Let us suppose that the force supplied to the bottom thread equals constant value f, starting from some moment of time. In dependence of the value of the force, one can observe break-up of either the bottom or the upper thread, or, alternatively, both threads can remain uncut. Assuming that the break occurs at the tension force T , indicate domains on the parameter plane (T, f) where all the mentioned situations take place. Mass of the ball if M. Assume that threads are characterized by constant spring coefficient k before the break-up. The threads are weightless.
1. A small box slides without friction along a curved surface with speed v approaching a bend with profile governed by a function f(x). Determine maximal and minimal values of speed of the box in the process of motion. Axis x is horizontal, the gravitational acceleration equals g.
2. On a horizontal plane with friction coefficient k a brick of mass m is placed. It is effected by a force F directed at some angle in respect to horizon. What is the minimal absolute value of the force that gives a possibility to shift the brick from its place? What is the optimal angle of the force direction?
3. Electrical charge Q is distributed uniformly over a ring of radius R. A little charged particle can move along the axis perpendicular to the plane of the ring and passing through its center. Where is the particle located at the moment of maximal force of its interaction with the ring?
4. Into a thick-wall glass of mass M an experimentalist pours gradually some liquid of density . Plot a qualitative dependency of the height of the mass center of the whole system versus a height of the liquid level h. What is the value of h that ensures a minimal height of the mass center? The inner part of the glass is a cylinder of cross-section area S and of height H. In the empty glass the mass center is placed on the distance l from the bottom.
5. It is known that in some carbon modifications and in carbon compounds the valency bonds of the carbon atoms are directed towards vertices of tetrahedron, and the angle between them equals 1090 28'. Prove that if you take a filled circle and cut a sector to obtain evolvent for a cone of maximal volume, the angle at the vertex of this cone will be equal to 1090 28'.
1. Calculate (1+i)2001.
2. With a help of the Euler formula derive trigonometric relations for , , , .
3. Using the Euler formula calculate the following sum: .
4. A particle moves along the circle of radius r with an angular velocity . Introduce a complex coordinate z and derive a relation for it as a function of time. Consider particular cases when the initial coordinates of the particle are defined as (r,0), (0,-r), . Derive a differential equation that determines the evolution of the complex coordinate.
5. What is the difference of two motions, and ?
6. A particle of charge e and mass m moves in the plane xy. Magnetic field B is directed perpendicularly to this plane. Derive a differential equation for the complex coordinate .
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