The author: Alexander P. Kuznetsov, Doctor of Sciences

- What is Physics?
- Numbers in Physics.
- Estimates of physical quantities. Examples of problems.
- Characteristic size.
- Scale.
- Accuracy in Physics.
- Mutual dependencies of physical quantities. Functions and graphics in Physics.
- Asymptotical behavior of quantities.
- Measurement and experiment in physics.
- Dimensions of physical quantities.
- Scaling as a way to derive a form of mutual dependence of physical quantities.
- Something about formulas.
- A picture for problem solving.
- Physical terminology.
- A Handbook as an assistant of a physicist.
- Discussing a problem.
- Scientific journals and a Library.
- Let us formulate a problem ourselves.

- Physical model.
- Algebra of approximate numbers.
- Geometry of different scales.
- Two time scales in one problem.
- Equations in physics.
- Qualitative theory. Examples of problems.
- Critical state.
- "Paradigm problems".

- Numerical sequences.
- Difference equations (Iterative maps).
- Transcendental equations.
- Derivative in physics.
- A tangent line.
- Maximum and minimum in physical problems. Examples of problems.
- Application of derivative in equation solving.
- Qualitative transformations and bifurcations.
- Exponential function.
- Integral.
- Differential equations.
- Complex numbers. Examples of problems.
- Level curves.

**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 *k*_{1}, and
that between the board and the surface is *k _{2}*. A
horizontally directed force

**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
109^{0} 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
109^{0} 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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