In the system shown in the figure, a block of mass M can slide along the rails without friction.
The load is retracted at an angle a from the vertical and released.
Determine the mass of the load m if the angle a does not change when the system moves.
A thin-walled gas-filled cylinder of mass M, height H, and base area S floats in water.
As a result of the loss of tightness in the lower part of the cylinder, the depth of its immersion increased by D H.
The atmospheric pressure is equal to P 0, the temperature does not change.
What was the initial gas pressure in the cylinder?
The closed metal chain is connected by a thread to the axis of the centrifugal machine and rotates with an angular velocity w.
In this case, the thread makes an angle a with the vertical.
Find the distance x from the center of gravity of the chain to the axis of rotation.
Inside a long tube filled with air, a piston is moved at a constant speed.
In this case, an elastic wave propagates in the pipe at a speed of S = 320 m / s.
Assuming the pressure drop at the wave propagation boundary equal to P = 1000 Pa, estimate the temperature drop.
Pressure in undisturbed air P 0 = 10 5 Pa, temperature T 0 = 300 K.
The figure shows two closed processes with the same ideal gas 1 - 2 - 3 - 1 and 3 - 2 - 4 - 2.
Determine in which of them the gas has done a great job.
Solutions to Olympiad problems in physics
Let T be the tension force of the thread, a 1 and a 2 - the accelerations of bodies with masses M and m.
Writing down the equations of motion for each of the bodies along the x axis, we obtain
a 1 M = T · (1-sina), a 2 m = T · sina.
Since the angle a does not change during motion, then a 2 = a 1 (1-sina). It is easy to see that
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From here
Considering the above, we finally find
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To solve this problem, it should be noted that
that the center of mass of the chain rotates along a circle of radius x.
In this case, only the gravity force applied to the center of mass and the thread tension T force act on the chain.
Obviously, centripetal acceleration can be provided only by the horizontal component of the thread tension force.
Therefore, mw 2 x = Tsina.
In the vertical direction, the sum of all forces acting on the chain is zero; then mg- Tcosa = 0.
From the obtained equations we find the answer
Let the wave move in the pipe at a constant velocity V.
Let us associate this value with a given pressure drop D P and a density difference D r in unperturbed air and a wave.
The pressure difference accelerates the "excess" of air with the density D r to the speed V.
Therefore, in accordance with Newton's second law, we can write
Dividing the last equation by the equation P 0 = R r T 0 / m, we obtain
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Since D r = D P / V 2, r = P 0 m / (RT), we finally find
A numerical estimate, taking into account the data given in the problem statement, gives the answer D T »0.48K.
To solve the problem, it is necessary to build graphs of circular processes in coordinates P-V,
since the area under the curve in such coordinates is equal to the work.
The result of this construction is shown in the figure.
Physics Olympiad
Grade 10
Physics Olympiads for grade 10
Olympiad problem in physics grade 10 (example):
Determine the refractive index of an unknown liquid inside a spherical flask, the position of the focus relative to the flask surface, and the radius of curvature of the flask.
Equipment. Spherical flask with liquid, laser, graph paper, tripod.
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| Rice. 1. |
Place the flask on the support. Attach the optical table to the stand and select the height of the stand so that the reflections of the laser beam from diametrically opposite sides lie in the same plane. If they are also combined with each other, then the laser beam will propagate along the diameter of the bulb (optical axis). To find the rear focal plane, let us select such a position of the graph paper, at which the laser spot on it does not move at small displacements of the laser in the direction perpendicular to the optical axis (Fig. 1). With the second strip of graph paper we measure the distance L from the bulb to the focal plane. Now let's shift the laser from the optical axis until the moment when the beam touches the edge of the bulb, then the laser displacement will coincide with the radius R of the bulb. Our setup turned out to be R ≈ L.
,
Physics Olympiad Problems for Grade 10 Students
Examples of Olympiad tasks Grade 10
Exercise 1.
Give a numerical estimate of the average number of water molecules,
evaporating from 1 cm 2 of its surface in 1 s during boiling.
An electric stove and a vessel with water are at your disposal.
What kind of measuring instruments do you need?
Task 2.
Two lead balls of the same mass are moving towards each other.
The speed of one of them is 3 times the speed of the other.
Determine the change in the temperature of the balls as a result of inelastic collision.
Task 3.
A balloon with helium at a pressure p 1 and a temperature T 1 has a mass M 1, and at a pressure p 2 and the same temperature it has a mass M 2. What mass of helium does the cylinder contain at pressure p and temperature T?
Task 4.
How to determine the specific heat of dissolution (melting) of table salt using a balance with weights, a thermometer, a vessel with water?
Task 5.
A ball was allowed to roll from the bottom up on the inclined board.
At a distance of 30 cm, from the start of the launch, the ball visited twice:
after 1 s and 2 s.
Determine the initial velocity of the ball and its acceleration.
Task 6.
A vessel with water at a temperature of 10 ° C was placed on an electric stove.
After 10 minutes, the water boiled.
How long will it take for the water to completely evaporate in the vessel?
Task 7.
Two identical small vessels with a volume of V = 0.03 m 3 each are connected by a horizontal tube,
the volume of which is equal to 2V, and the section is 0.1 m 2.
There is a thin piston in the middle of the tube,
able to move without friction.
The pressure in the vessels is p.
To one of the vessels by means of a tube of negligible volume, a third vessel of exactly the same type was connected, the gas pressure in which is equal to 2p.
Determine the movement of the piston after equilibrium has been established.
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School round of the Olympiad
Physics
Grade 10
Question
Answer
PART 1
For each of tasks 1-10, 4 answer options are given, of which only one is correct. The number of the correct answer must be entered into the table.
1. The graph shows the dependence of the speed of a rectilinearly moving body on time.
Determine the acceleration modulus of the body.
2. The crane lifts the load with constant acceleration. A force equal to 8⋅ 10 3 N acts on the load from the cable side. The force acting on the cable from the load side,
1) is equal to 8 ⋅ 10 3 N and is directed downward
2) less than 8 ⋅ 10 3 N and directed downward
3) more than 8 ⋅ 10 3 N and directed up
4) is equal to 8 ⋅ 10 3 N and is directed upwards
3. A stone weighing 200 g is thrown at an angle of 45 ° to the horizon with an initial speed υ = 15 m / s. The modulus of the force of gravity acting on the stone at the moment of throwing is equal to
4
... The balls move at the speeds shown in the figure and stick together when they collide. How will the impulse of the balls be directed after the collision?
5
... To destroy the obstacle, a massive ball is often used swinging on the boom of a crane (see figure). What energy transformations occur when the ball moves from position A to position B?
1) the kinetic energy of the ball is converted into its potential energy
2) the potential energy of the ball is converted into its kinetic energy
3) the internal energy of the ball is converted into its kinetic energy
4) the potential energy of the ball is completely converted into its internal
6
... The graph shows the results of measuring the voltage at the ends of the section AB a direct current circuit consisting of two series-connected resistors, at different values of the resistance of the resistor R 2 and constant current I(see figure).
WITH 
taking into account measurement errors (Δ R= ± 1 Ohm, Δ U= ± 0.2 V) find the expected voltage at the ends of the circuit section AB at R 2 = 50 ohms.
7. A current I flows through a conductor with resistance R. How will the amount of heat released in the conductor per unit of time change if its resistance is doubled and the current is reduced by 2 times?
1) will increase 2 times
2) will decrease by 2 times
3) will not change
4) decrease by 8 times
8. The weight suspended on the thread performs harmonic vibrations. The table shows the coordinates of the weight at regular intervals. What is the approximate maximum speed of the weight?
PART 2
The answer to the task of this part (problem 9) is a sequence of numbers, which you will enter in the table of answers.
NS 
arik is thrown vertically upward with an initial speed 
(see figure). Establish a correspondence between the graphs and physical quantities, the dependence of which on time these graphs can represent (t 0 - flight time). For each position of the first column, select the corresponding position of the second and write down the selected numbers in the table under the corresponding letters.
PHYSICAL QUANTITIES
1) the y coordinate of the ball
2) the projection of the ball velocity υ y
3) the projection of the ball acceleration a y
4) the projection F y of the force of gravity acting on the ball
Part 3
The complete correct solution to each of the problems 10-11 should include laws and formulas, the use of which is necessary and sufficient to solve the problem, as well as mathematical transformations, calculations with a numerical answer and, if necessary, a figure explaining the solution.
10. It is necessary to melt ice weighing 0.2 kg and having a temperature of 0 ºС. Is this task feasible if the power consumption of the heating element is 400 W, the heat loss is 30%, and the operating time of the heater should not exceed 5 minutes?
1
1. Loads of masses M = 1 kg and m are connected by a light inextensible thread thrown over a block, along which the thread can slide without friction (see figure). A load of mass M is located on a rough inclined plane (the angle of inclination of the plane to the horizon α = 30 °, coefficient of friction μ = 0.3). What is the maximum value of the mass m, at which the system of loads still does not leave the initial state of rest? Explain the solution with a schematic drawing indicating the forces used.
Solutions
Objective 1.
A grenade, thrown vertically upward, exploded at the top point into many identical fragments flying at the same speed of 20 m / s. Determine the time interval during which the fragments fell to the ground.
(10 points)
| Possible Solution | |
| Let t 1 (t 2) be the time of movement of a fragment flying vertically down (vertically up). Let us write down the equations of motion of the fragments: 0 = H - ʋ 0 t 1 - (1); 0 = Н + ʋ 0 t 2 - (2) Analysis of the movement of the fragments leads to the conclusion: a fragment flying vertically downward (t 1) will fall to the ground before anyone else. More time will be spent on the fall of a fragment flying t 2. Then the required time Δt = t 2 - t 1; Solving equations (1) and (2) together, we obtain: Δt = t 2 - t 1 = 4 s. | |
| Points | |
| all newly introduced letter designations of physical quantities are described (time of movement of fragments, time interval) the equations of motion of the fragments of motion are written in general form for the first shard 0 = H - ʋ 0 t 1 - for the second shard 0 = H + ʋ 0 t 2 - ; more time will be spent on falling a fragment flying t 2; required time Δt = t 2 - t 1 ; ; Δt = 4 s. |
Objective 2.
A bucket containing m = 10 kg of a mixture of ice water was brought into the room and the temperature of the mixture was measured immediately. The graph of temperature versus time t (t) is shown in the figure. How much ice was in the bucket when they brought it into the room? Specific heat of water c = 4200 J / (kg o C), specific heat of melting of ice l = 330 kJ / kg. Disregard the heat capacity of the bucket.
(10 points)
| Possible Solution | |
Ice melting in a bucket and water heating occurs due to heat exchange with the environment. Since the increase in temperature from time to time in the range under consideration is linear, the power P of the heat flux can be considered constant. The heat balance equation for ice melting is m l l = Pt 0, where m l is the mass of ice in a bucket, t 0 = 50 min is the time of ice melting. The heat balance equation during water heating is mсΔt = РΔt, where Δt is the water heating time. From the graph we determine  ... Thus  |
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| Evaluation criteria for the assignment | Points |
| The complete correct solution is given, including the following elements: ; a complete correct explanation is presented, indicating the observed phenomena and laws: they explained that the melting of ice in a bucket and the heating of water occurs due to heat exchange with the environment; We noticed that the increase in temperature from time to time in the range under consideration is linear, therefore, the power P of the heat flux can be considered a constant medium, the heat balance equation for ice melting is written m l l = Pt 0 ;
heat balance equation for water heating mсΔt = РΔt ;
define the necessary mathematical transformations and calculations were carried out, leading to the correct numerical answer;
the correct answer is presented, indicating the units of measurement of the desired value
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Objective 3.
Resistors with resistances R 1 = 1 kOhm, R 2 = 2 kOhm, R 3 = 3 kOhm, R 4 = 4 kOhm are connected to a constant voltage source U 0 = 33V through terminals A and B. Two ideal ammeters A 1, A were connected to the resistors 2. Determine the readings of the ammeters I 1, I 2.
Points).
| Possible Solution | |||
Let us determine the currents I i flowing through the resistors R i (i = 1, 2, 3, 4). Since ammeters are ideal, an equivalent electrical circuit can be considered. For this circuit, R AB = RAC + RCB =. Total current in the circuit  To determine the ammeter readings, we write down the law of conservation of currents at nodes d and c (the selected direction of currents is shown in the figure): I 1 = I R 1 - I R 3 = 5 mA, I 2 = I R 3 - I R 4 = 4 mA   |
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| Evaluation criteria for the assignment | Points | ||
| The complete correct solution is given, including the following elements: Explanatory drawing made; the necessary mathematical transformations and calculations were carried out, leading to the correct numerical answer determined the resistance R AC; determined the resistance R CB; determined the resistance R AB; | |||
| determined I 0; determined I R 1; determined I R 2; determined I R 3; the correct answer is presented, indicating the units of measurement of the desired value: I 1 = 5 mA, I 2 = 4 mA | |||
Task 4.
A piece of ice is tied with a thread to the bottom of a cylindrical vessel with water (see figure). There is a certain volume of ice above the surface of the water. The thread is stretched with a force of T = 1N. How much and how will the water level in the vessel change if the ice melts? The area of the bottom of the vessel is S = 400 cm 2, the density of water is ρ = 1 g / cm 3.
(10 points)
| Possible Solution | |
| Let us write down the condition for a piece of ice to float in water: m l g + T = F A = ρ in V p.h. g; where V p.h is the volume of a piece of ice submerged in water. Let's find the initial water level in the vessel (1), where V о is the initial volume of water in the vessel before the ice melts. Accordingly (2), where h 2 is the water level in the vessel after the ice has melted, V 1 is the volume of water obtained from the ice. Solving (1) and (2) together, we obtain h 1 –h 2 = (V p.p. –V 1) / S; we find V p.h = (m л g + Т) / (ρ in. g). Let us take into account m l = m 1, where m 1 is the mass of water obtained from ice m 1 = ρ in V 1; V 1 = m l / ρ v. Then h 1 –h 2 = ((m l g + T) / ρ in g. - m l / ρ in) / S = 2.5 mm | |
| Evaluation criteria for the assignment | Points |
| The complete correct solution is given, including the following elements: an explanatory drawing was made, indicating all the forces acting; all newly introduced letter designations of physical quantities are described; the complete correct explanation is presented, indicating the observed phenomena and laws: the condition for a piece of ice to float in water is written: m l g + T = F A = ρ in V p.h. g; wrote down the formula for calculating h 1; wrote down the formula for calculating h 2; the necessary mathematical transformations and calculations were carried out, leading to the correct numerical answer: h 1 –h 2 = (V p.p. –V 1) / S; V p.h = (m l g + T) / (ρ in. G); V 1 = m l / ρ in; h 1 –h 2 = ((m l g + T) / ρ in g. - m l / ρ in) / S. The correct answer is presented, indicating the units of measurement of the desired value: h 1 –h 2 = 2.5 mm |
1 .From the same point vertically upward with a time interval Δt, two balls are thrown with a speed V. The balls move along one straight line in the field of gravity. How long after the launch of the second balloon will they collide?
Solution. Let's write down the equation of coordinates of the first and second bodies when moving vertically upward. At the point of intersection of the trajectories, the coordinates of the bodies are equal to y 1 = y 2. (2b) Therefore, we equate these two equations and solve for the unknown value t. 
2. A stone of mass m = 100 g is thrown horizontally from the top of a hill, the slope of which makes an angle of 30 ° with the horizon. Determine what kind of work was completed during the throw if the stone fell on the slope at a distance of 40 m from the top. Consider that the throw was made directly from the ground. Neglect air resistance.
Solution: Let's introduce the coordinate system as shown in the figure. Let us denote by V 0 the initial velocity of the stone. The kinematic equations of motion are: 
, therefore, the equation of its trajectory is. The equation of the inclined plane of the hill surface is:. At the point where the stone falls, has a coordinate, the equality 
... It turns out: 
... Work perfect when thrown: looks like this 
3.
A cart with a mass of M = 500 g, located on the table, is fastened with a thread thrown over a block (the block is attached to the right edge of the table) with a load of m = 200 g. At the initial moment, the cart had a speed of V 0 = 7 m / s and moved to the left along a horizontal plane ...
Define:
a) the magnitude and direction of the trolley speed;
b) the place where she will be, and the path she traveled through t = 5 s.
(Acceleration of gravity g = 9.8 m / s2).
Solution: According to Newton's second law: 
Before stopping 
... The speed at the moment of stopping. Subsequent movement time 
... After stopping, the path will pass: taking into account that the movement against the Ox axis 
... After five seconds, the cart will be in its starting position. 
Answer: 17.5m; 7m / s; in the starting place.
4. The electric motor of the machine is driven from a network with a voltage of U = 220 V. When the machine is operating, the current flowing through the motor is I = 11 A. What part of the consumed energy is converted into mechanical work if the resistance of the motor winding is R = 5 Ohm?
Solution:
(1); where 
... Substitute in formula (1): 
; substitute the numerical values: 
Answer: 3/4 part (or 75%) of the spent energy turned into mechanical work.
