A string with a load suspended on it is deflected at an angle α and let go. At what angle β will the thread with the load deviate if, during its movement, it is delayed by a pin placed on the vertical, in the middle of the length of the thread?
Answer
β = arccos(2cos α -1).
1. A body is thrown vertically upward with a speed v 0 = 16 m/s. At what height h Is the kinetic energy of a body equal to its potential energy?
2. With what initial speed should the ball be thrown from a height h so that he jumps to height 2 h? The impact is elastic. Ignore air resistance.
Answer
1. h≈ 6.5 m.
From a tower high H= 25 m a stone is thrown horizontally with a speed v 0 = 15 m/s. Find the kinetic ( K) and potential ( U) energy of the stone one second after the start of movement. Mass of stone m= 0.2 kg. Ignore air resistance.
Answer
K= 32.2 J; U= 39.4 J.
Determine the amount of kinetic energy K a body of mass 1 kg, thrown horizontally with a speed of 20 m/s, at the end of the fourth second of its motion. Accept g\u003d 10 m / s 2.
Answer
K= 1000 J.
Flexible uniform rope length L lies on a smooth horizontal table. One end of the rope is at the edge of the table. At some point, from a small push, the rope began to move, continuously sliding off the table. How does the acceleration and speed of the rope depend on the length X piece of it hanging from the table? What will be the speed of the rope by the time it slides off the table?
Answer
a = xg/L; ; .
Rope length L passed over the pin. At the initial moment, the ends of the rope were at the same level. After a slight push, the rope began to move. Determine speed v rope by the time it slips off the pin. Ignore friction.
Answer
Skater accelerating to speed v= 27 km/h, enters an ice mountain. To what height H from the initial level, a speed skater will enter with acceleration if the rise of the mountain is h= 0.5 m for each s\u003d 10 m horizontally and the coefficient of friction of skates on ice k = 0,02?
Answer
H≈ 2 m.
body mass m= 1.5 kg thrown vertically upward from a height h= 4.9 m with speed v 0 = 6 m/s, fell to the ground with a speed v= 5 m/s. Determine the work of air resistance forces.
Answer
A≈ -80.2 J.
A stone of mass 50 g, thrown at an angle to the horizon from a height of 20 m above the earth's surface with a speed of 18 m/s, fell to the ground with a speed of 24 m/s. Find work to overcome the forces of air resistance.
Answer
A≈ 3.5 J.
Aircraft mass m= 10 3 kg flies horizontally at a height H= 1200 m with speed v 1 = 50 m/s. Then the engine is turned off, the plane goes into a gliding flight and reaches the ground at a speed v 2 = 25 m/s. Determine the average force of air resistance during the descent, assuming the length of the descent is 8 km.
Answer
F cf ≈ 1570 N.
body mass m\u003d 1 kg moves along the table, having a speed at the starting point v 0 = 2 m/s. Reaching the edge of the table, the height of which h= 1 m, the body falls. Coefficient of friction of the body on the table k= 0.1. Determine the amount of heat released during an inelastic impact on the ground. The path traveled by the body on the table s= 2 m.
Answer
Q≈ 9.8 J.
A load attached to a vertical spring is slowly lowered to the equilibrium position, and the spring is stretched to a length X 0 . How much will the spring stretch if the same weight is allowed to fall freely from a position where the spring is not stretched? Which top speed v max will the load reach? What is the nature of the cargo movement? Load weight m. Ignore the mass of the spring.
Answer
2x 0; ; oscillatory nature of the movement of the load.
Falling from a height h\u003d 1.2 m, the pile is driven with a load, which, from the impact, goes into the ground for s\u003d 2 cm. Determine the average impact force F Wed and its duration τ if the weight of the load M= 5·10 2 kg, the mass of the pile is much less than the mass of the load.
Answer
F cf ≈ 3 10 5 N; τ ≈ 8 10 -3 s.
From a mountain high h= 2 m and base b= 5 m the sled leaves, which then stops after passing the horizontal path l= 35 m from the base of the mountain. Find the coefficient of friction.
Answer
k = 0,05.
Steel ball mass m= 20 g, falling from a height h 1 = 1 m on a steel plate, bounces off it to a height h 2 \u003d 81 cm. Find: a) the impulse of the force acting on the plate during the impact; b) the amount of heat released during the impact.
Answer
but) p= 0.17 N s;
b) Q= 3.7 10 -2 J.
A light ball begins to fall freely and, having flown a distance l, collides elastically with a heavy plate moving up at a speed u. To what height h will the ball bounce after being hit?
Answer
Balloon, held by a rope, rose to a certain height. How has the potential energy of the ball-air-Earth system changed?
Answer
The potential energy of the ball-air-Earth system has decreased, since when the ball rises up, the volume occupied by the ball is replaced by air having a mass, b about bigger than a ball.
Hockey puck having initial speed v 0 = 5 m/s, slides before hitting the side of the site s= 10 m. The impact is assumed to be absolutely elastic, the friction coefficient of the puck on ice k= 0.1, air resistance is neglected. Determine which path l the puck will pass after the impact.
Answer
l≈ 2.7 m.
The body slides without friction from a wedge lying on a horizontal plane two times: the first time the clip is fixed; the second time the wedge can slide without friction. Will the speed of the body at the end of sliding off the wedge be the same in both cases if the body slides from the same height both times?
Answer
The speed of the body in the first case is greater than in the second.
Why is it difficult to jump to the shore from a light boat standing close to the shore, and easy to do from a steamer located at the same distance from the shore?
Answer
When jumping from a steamer, a person does less work than when jumping from a boat.
mass skater M\u003d 70 kg, standing on skates on ice, throws a stone in a horizontal direction with a mass m= 3 kg with speed v= 8 m/s relative to the Earth. Find how far s the skater will roll back if the coefficient of friction of the skates on the ice k = 0,02.
Answer
s≈ 0.29 m.
A man stands on a stationary cart and throws horizontally a stone with a mass m= 8 kg with speed v 1 = 5 m/s relative to the Earth. Determine what kind of work the person does in this case, if the mass of the trolley together with the person M= 160 kg. Analyze the dependence of work on mass M. Ignore friction.
Answer
A≈ 105 J.
Mass rifle M= 3 kg suspended horizontally on two parallel threads. When fired as a result of recoil, she deviated upwards by h= 19.6 cm.
bullet weight m= 10 g. Determine the speed v 1 from which the bullet was fired.
Answer
v 1 ≈ 590 m/s.
A bullet traveling horizontally at a speed v\u003d 40 m / s, hits a bar suspended on a thread of length l= 4 m, and gets stuck in it. Determine Angle α , by which the bar will deviate if the mass of the bullet m 1 \u003d 20 g, and a bar m 2 = 5 kg.
Answer
α ≈ 15º.
A bullet flying horizontally hits a ball suspended from a very light rigid rod and gets stuck in it. Bullet weight in n= 1000 times less than the mass of the ball. Distance from the suspension point of the rod to the center of the ball l= 1 m. Find the speed of the bullet v, if it is known that the rod with the ball deviated from the impact of the bullet by an angle α = 10°.
Answer
v≈ 550 m/s.
bullet mass m 1 = 10 g, flying horizontally with a speed v 1 \u003d 600 m / s, hit a wooden block freely suspended on a long thread with a mass m 2 = 0.5 kg and got stuck in it, going deep into s= 10 cm Find strength F with the resistance of the tree to the movement of the bullet. To what depth S 1 a bullet will enter if the same block is fixed.
Answer
F c ≈ 1.8 10 4 N; s 1 ≈ 10.2 cm.
into a ball at rest with mass M\u003d 1 kg, suspended on a long rigid rod, fixed in a suspension on a hinge, a bullet of mass m= 0.01 kg. The angle between the direction of flight of the bullet and the line of the rod is equal to α = 45°. Center punch. After the impact, the bullet gets stuck in the ball and the ball, together with the bullet, deviates, rises to a height h= 0.12 m relative to the initial position. Find bullet speed v. Ignore the mass of the rod.
Answer
v≈ 219 m/s.
The pendulum is a straight thin rod of length l= 1.5 m, at the end of which there is a steel ball of mass M= 1 kg. The ball is hit by one flying horizontally with a speed v= 50 m/s steel ball mass m= 20 g. Determine the angle of maximum deflection of the pendulum, considering the impact to be elastic and central. Ignore the mass of the rod.
Answer
α ≈ 30º.
Two weights of unequal masses are suspended on a thread thrown over a block m 1 and m 2. Find the acceleration of the center of mass of this system. Solve the problem in two ways, applying: 1) the law of conservation of energy and 2) the law of motion of the center of mass. Ignore the masses of the block and thread.
Answer
.
Hammer mass m= 1.5 t hits a hot ingot lying on an anvil and deforms it. The mass of the anvil together with the blank M\u003d 20 tons. Determine the efficiency η when struck by a hammer, assuming the blow is inelastic. Consider the work done during the deformation of the blank as useful.
Answer
η ≈ 93 %.
body mass m 1 hits inelastically on a body at rest with a mass m 2. Find a share q the kinetic energy lost in the process.
Answer
q = m 2 /(m 1 +m 2).
On the front edge of the platform M moving horizontally without friction with a speed v, lower from a small height a short bar with a mass m. What is the minimum platform length l the block will not fall from it if the coefficient of friction between the block and the platform k. How much heat Q will stand out.
Answer
; 
.
body mass m= 1 kg, lying on a long horizontal platform of a cart at rest, report the speed v= 10 m/s. Coefficient of friction of the body on the platform k= 0.2. How far will the cart travel by the time the body stops on it? How much heat will be released when the body moves along the platform? The cart rolls along the rails without friction, its mass is M= 100 kg.
Answer
s≈ 0.25 m; Q≈ 50 J.
Two loads of masses m 1 = 10 kg and m 2 = 15 kg suspended on threads long l= 2 m so that they touch each other. The smaller load was rejected by an angle α = 60° and released. To what height will both weights rise after the impact? The impact of the loads is assumed to be inelastic. How much heat is released in this case?
Answer
h≈ 0.16 m; Q≈ 58.8 J.
The ball moves between two very heavy vertical parallel walls, colliding with them according to the law of perfectly elastic impact. One of the walls is fixed, the other moves away from it at a constant horizontal speed u x = 0.5 m/s. Determine the number of collisions and and the final speed v x of the ball, if before the first collision with the wall it was equal to v 0x = 19.5 m/s.
Answer
Number of collisions n = 19; v x = 0.5 m/s.
Two balls are suspended on parallel threads of the same length so that they are in contact. Masses of balls m 1 = 0.2 kg and m 2 \u003d 100 g. The first ball is deflected so that its center of gravity rises to a height h= 4.5 cm, and let go. To what height will the balls rise after the collision, if the impact: a) is elastic; b) inelastic?
Answer
but) h 1 \u003d 5 10 -3 m, h 2 = 0.08 m;
b) H= 0.02 m.
How many times will the speed of a helium atom decrease after a central elastic collision with a stationary hydrogen atom, whose mass is four times less than the mass of a helium atom?
Answer
n = 5/3.
A ball lying on a smooth horizontal surface is hit by another ball of the same radius moving horizontally. An elastic central impact occurs between the balls. Plot the dependence of the share of transferred energy on the ratio of the masses of the balls α =m 1 /m 2 .
Answer
.
To obtain slow neutrons, they are passed through substances containing hydrogen (for example, paraffin). Find what is the largest part of its kinetic energy of a neutron with a mass m 0 can transfer: a) proton (mass m 0); b) the nucleus of the lead atom (mass m = 207 m 0). The largest part of the transferred energy corresponds to the elastic central impact.
Answer
a) 100%, with an elastic collision of particles with the same mass, an exchange of velocities occurs;
Two perfectly elastic balls with masses m 1 and m 2 are moving along the same straight line with velocities v 1 and v 2. During the collision, the balls begin to deform and part of the kinetic energy is converted into potential energy of deformation. Then the deformation decreases, and the stored potential energy again transforms into kinetic energy. Find the value of the maximum potential energy of deformation.
Answer
.
A small streamlined body with density ρ 1 falls in the air from a height h on the surface of a liquid with a density ρ 2 , and ρ 1 < ρ 2. Determine Depth h 1 body immersion in liquid, immersion time t and acceleration a. Ignore the fluid resistance.
Answer
; 
; .
On a thread long l suspended load of mass m. Determine the minimum height to which this load must be raised so that, falling, it breaks the thread, if the minimum load of mass M, suspended on a thread and breaking it, stretches the thread at the moment of breaking by 1% of its length. Assume that Hooke's law is valid for the thread up to the break.
Answer
h min = 0.01 ml/(2m).
Determine the maximum range of the jet s from a syringe with a diameter d\u003d 4 cm, on the piston of which the force presses F= 30 N. Liquid density ρ \u003d 1000 kg / m 3. Neglect air resistance S rep ≪ S Porsche).
Answer
s≈ 4.9 m.
Cylinder diameter D filled with water and placed horizontally. At what speed u the piston moves in the cylinder if a force acts on it F, and a jet with a diameter d? Ignore friction. Gravity is ignored. Liquid Density ρ .
Answer
.
On a smooth horizontal wire ring, two beads can slide without friction in masses m 1 and m 2. Initially, the beads were connected by a thread and between them was a compressed spring. The thread is burned. After the beads have begun to move, the spring is removed. Where will the beads collide for the 11th time? Collisions of beads are absolutely elastic. Ignore the mass of the spring.
Answer
l 1 /l 2 = m 2 /m 1 , where l 1 and l 2 are the lengths of the arcs of the ring from the point of the beginning of the movement to the point of the 11th collision.
Proton mass m, flying at a speed v 0 , collided with an immobile atom of mass M, after which he began to move in the opposite direction at a speed of 0.5 v o , and the atom has passed into an excited state. Find speed v and energy E excitation of the atom.
Answer
; 
.
During the decay of an immobile nucleus, three fragments are formed with masses m 1, m 2 and m 3 with a total kinetic energy E 0. Find the velocities of the fragments if the directions of the velocities make angles of 120° with each other.
Answer
;
;
;
IN general view:
Another similar ball hits a stationary ball not along the line of centers. At what angle α Will the balls scatter if they are absolutely elastic and absolutely smooth?
Answer
α = 90º.
two balls BUT And IN with various unknown masses elastically collide with each other. Ball BUT before the collision was at rest, and the ball IN moving at a speed v. After hitting the ball IN acquired speed 0.5 v and began to move at right angles to the direction of its original movement. Determine the direction of the ball BUT and his speed v A after the collision.
Answer
v A=0.66 v.
When bombarded with helium α - particles with energy E 0 the incident particle deviated by an angle φ = 60° with respect to the direction of its movement before the collision. Assuming the impact to be absolutely elastic, determine the energies α -particles Wα and nuclei W He after the collision. The energy of thermal motion of helium atoms is much less E 0 .
Answer
Wα = 1/4 E 0 ; W He = 3/4 E 0 .
A smooth ball of soft lead collides with a similar ball, which is initially at rest. After the collision, the second ball flies at an angle α to the direction of the speed of the first ball before the collision. Determine Angle β , under which the balls fly apart after the collision. What part of the kinetic energy T will turn into heat upon collision Q?
Answer
β = arctg(2tg α ); Q/T= ½ cos 2 α .
Ball mass m, moving at a speed v, hits a ball at rest with mass m/2 and after elastic impact continues to move at an angle α = 30° to the direction of its initial movement. Find the speed of the balls after the collision.
Energy and momentum are the most important concepts in physics. It turns out that conservation laws play an important role in nature in general. The search for conserved quantities and the laws from which they can be obtained is the subject of research in many branches of physics. Let us derive these laws in the simplest way from Newton's second law.
Law of conservation of momentum.Pulse, or amount of movementp defined as the product of the mass mmaterial point for speed V: p= mV. Newton's second law, using the definition of momentum, is written as
= dp= F, (1.3.1)
here F is the resultant of the forces applied to the body.
closed system called a system in which the sum of external forces acting on the body is equal to zero:
F= å Fi= 0 . (1.3.2)
Then the change in the momentum of the body in a closed system according to Newton's second law (1.3.1), (1.3.2) is
dp= 0 . (1.3.3)
In this case, the momentum of the particle system remains constant value:
p= å pi= const . (1.3.4)
This expression is law of conservation of momentum, which is formulated as follows: when the sum of external forces acting on a body or system of bodies is equal to zero, the momentum of the body or system of bodies is a constant value.
Law of energy conservation. In everyday life, by the concept of "work" we understand any useful work of a person. In physics, it is studied mechanical work, which occurs only when the body moves under the action of a force. Mechanical work ∆A is defined as the scalar product of the force F applied to the body, and body displacement Δ r as a result of this force:
A A= (F, Δ r) = F A r cosα. (1.3.5)
In formula (1.3.5), the sign of work is determined by the sign of cos α.
Wanting to move the cabinet, we press on it with force, but if it does not move at the same time, then mechanical work we don't do. One can imagine the case when the body moves without the participation of forces (by inertia),
in this case no mechanical work is done either. If a system of bodies can do work, then it has energy.
Energy is one of the most important concepts not only in mechanics, but also in other areas of physics: thermodynamics and molecular physics, electricity, optics, atomic, nuclear and particle physics.
In any system belonging to the physical world, energy is conserved in any process. Only the form into which it passes can change. For example, when a bullet hits a brick, part of the kinetic energy (moreover, more) is converted into heat. The reason for this is the presence of a frictional force between the bullet and the brick, in which it moves with great friction. When the turbine rotor rotates, mechanical energy is converted into electrical energy, and at the same time, a current appears in a closed circuit. Energy released during combustion chemical fuel, i.e. the energy of molecular bonds is converted into thermal energy. The nature of chemical energy is the energy of intermolecular and interatomic bonds, essentially representing molecular or atomic energy.
Energy is a scalar quantity that characterizes the ability of a body to do work:
E2-E1= ∆A. (1.3.6)
When mechanical work is performed, the energy of a body changes from one form to another. The energy of a body can be in the form of kinetic or potential energy.
energy mechanical movement
W kin = .
called kinetic energy forward movement of the body. Work and energy in the SI system of units are measured in joules (J).
Energy can be determined not only by the motion of bodies, but also by their mutual arrangement and form. This energy is called potential.
Potential energy is possessed relative to each other by two loads connected by a spring, or by a body located at a certain height above the Earth. This last example refers to the gravitational potential energy when a body moves from one height above the Earth to another. It is calculated according to the formula
E full \u003d E kin + U
E kin \u003d mv 2 / 2 + Jw 2 / 2 - kinetic energy of translational and rotary motion,
U = mgh is the potential energy of a body of mass m at a height h above the Earth's surface.
F tr \u003d kN - sliding friction force, N - normal pressure force, k - friction coefficient.
In the case of an off-center impact, the law of conservation of momentum
S p i= const is written in projections on the coordinate axes.
The law of conservation of angular momentum and the law of dynamics of rotational motion
S L i= const is the law of conservation of angular momentum,
L OS \u003d Jw - axial angular momentum,
L orb = [ rp] is the orbital angular momentum,
dL/dt=SM ext - the law of rotational motion dynamics,
M= [RF] = rFsina – moment of force, F – force, a – angle between radius-vector and force.
A \u003d òMdj - work during rotational motion.
Mechanics section
Kinematics
A task
A task. The dependence of the path traveled by the body on time is given by the equation s = A–Bt+Ct 2 . Find the speed and acceleration of the body at time t.
Solution example
v \u003d ds / dt \u003d -B + 2Ct, a \u003d dv / dt \u003d ds 2 / dt 2 \u003d 2C.
Options
1.1. The dependence of the path traveled by the body on time is given by
the equation s \u003d A + Bt + Ct 2, where A \u003d 3m, B \u003d 2 m / s, C \u003d 1 m / s 2.
Find the speed in the third second.
2.1. The dependence of the path traveled by the body on time is given by
the equation s \u003d A + Bt + Ct 2 + Dt 3, where C \u003d 0.14m / s 2 and D \u003d 0.01 v / c 3.
After how much time after the start of motion, the acceleration of the body
will be equal to 1 m / s 2.
3.1. The wheel, rotating uniformly accelerated, has reached the angular velocity
20 rad/s through N = 10 revolutions after the start of motion. To find
angular acceleration wheels.
4.1. A wheel with a radius of 0.1 m rotates so that the dependence of the angle
j \u003d A + Bt + Ct 3, where B \u003d 2 rad / s and C \u003d 1 rad / s 3. For points lying
on the wheel rim, find after 2 s after the start of movement:
1) angular velocity, 2) linear velocity, 3) angular
acceleration, 4) tangential acceleration.
5.1. A wheel with a radius of 5 cm rotates so that the dependence of the angle
rotation of the wheel radius versus time is given by the equation
j \u003d A + Bt + Ct 2 + Dt 3, where D \u003d 1 rad / s 3. Find for points lying
on the wheel rim, the change in tangential acceleration for
every second of movement.
6.1. A wheel with a radius of 10 cm rotates so that the dependence
linear velocity of the points lying on the wheel rim, from
time is given by the equation v \u003d At + Bt 2, where A \u003d 3 cm / s 2 and
B \u003d 1 cm / s 3. Find the angle formed by the vector of the complete
acceleration with wheel radius at time t = 5s after
start of movement.
7.1. The wheel rotates so that the dependence of the angle of rotation of the radius
wheel versus time is given by the equation j =A +Bt +Ct 2 +Dt 3 , where
B \u003d 1 rad / s, C \u003d 1 rad / s 2, D \u003d 1 rad / s 3. Find the radius of the wheel,
if it is known that by the end of the second second of motion
the normal acceleration of the points lying on the wheel rim is
and n \u003d 346 m / s 2.
8.1. The radius vector of a material point changes with time according to
law R=t 3 I+ t2 j. Determine for the moment of time t = 1 s:
speed module and acceleration module.
9.1. The radius vector of a material point changes with time according to
law R=4t2 I+ 3t j+2to. Write an expression for a vector
speed and acceleration. Determine for time t = 2 s
speed module.
10.1. A point moves in the xy plane from a position with coordinates
x 1 = y 1 = 0 with speed v= A i+Bx j. Define Equation
the trajectory of the point y(x) and the shape of the trajectory.
Moment of inertia
distance L/3 from the beginning of the rod.
Solution example.
M - rod mass J = J st + J gr
L - rod length J st1 \u003d mL 2 / 12 - rod moment of inertia
2m is the mass of the weight relative to its center. By theorem
Steiner find the moment of inertia
J=? rod relative to the o-axis, spaced from the center by a distance a = L/2 - L/3 = L/6.
J st \u003d mL 2 / 12 + m (L / 6) 2 \u003d mL 2 / 9.
According to the principle of superposition
J \u003d mL 2 / 9 + 2m (2L / 3) 2 \u003d mL 2.
Options
1.2. Determine the moment of inertia of a rod with a mass of 2m relative to an axis spaced from the beginning of the rod by a distance L/4. At the end of the rod, the concentrated mass m.
2.2. Determine the moment of inertia of the rod with mass m relative to
axis spaced from the beginning of the rod at a distance L / 5. At the end
rod concentrated mass 2m.
3.2. Determine the moment of inertia of a rod with a mass of 2m relative to an axis spaced from the beginning of the rod by a distance L/6. At the end of the rod, the concentrated mass m.
4.2. Determine the moment of inertia of a rod with a mass of 3m relative to an axis spaced from the beginning of the rod by a distance L/8. At the end of the rod, the concentrated mass is 2m.
5.2. Determine the moment of inertia of a rod with a mass of 2m about the axis passing through the beginning of the rod. Concentrated masses m are attached to the end and middle of the rod.
6.2. Determine the moment of inertia of a rod with a mass of 2m about the axis passing through the beginning of the rod. A concentrated mass 2m is attached to the end of the rod, and a concentrated mass 2m is attached to the middle.
7.2. Determine the moment of inertia of the rod with mass m about the axis, which is L/4 from the beginning of the rod. Concentrated masses m are attached to the end and middle of the rod.
8.2. Find the moment of inertia of a thin homogeneous ring of mass m and radius r about an axis lying in the plane of the ring and spaced from its center by r/2.
9.2. Find the moment of inertia of a thin homogeneous disk of mass m and radius r about an axis lying in the plane of the disk and spaced from its center by r/2.
10.2. Find the moment of inertia of a homogeneous ball of mass m and radius
r relative to the axis spaced from its center by r/2.
I will start with a couple of definitions, without the knowledge of which further consideration of the issue will be meaningless.
The resistance that a body exerts when trying to set it in motion or change its speed is called inertia.
Measure of inertia - weight.
Thus, the following conclusions can be drawn:
- The greater the mass of the body, the more it resists the forces that try to bring it out of rest.
- The greater the mass of the body, the more it resists the forces that try to change its speed if the body moves uniformly.
Summarizing, we can say that the inertia of the body counteracts attempts to give the body acceleration. And the mass serves as an indicator of the level of inertia. The greater the mass, the greater the force must be applied to influence the body in order to give it acceleration.
Closed system (isolated)- a system of bodies that is not influenced by other bodies that are not included in this system. Bodies in such a system interact only with each other.
If at least one of the two conditions above is not met, then the system cannot be called closed. Let there be a system consisting of two material points with velocities and respectively. Imagine that there was an interaction between the points, as a result of which the speeds of the points changed. Denote by and the increments of these velocities during the time of interaction between the points . We will assume that the increments have opposite directions and are related by the relation 
. We know that the coefficients and do not depend on the nature of the interaction of material points - this is confirmed by many experiments. The coefficients and are characteristics of the points themselves. These coefficients are called masses (inertial masses). The given relationship for the increment of velocities and masses can be described as follows.
The ratio of the masses of two material points is equal to the ratio of the increments of the velocities of these material points as a result of the interaction between them.
The above relation can be presented in another form. Let us denote the speeds of the bodies before the interaction as and respectively, and after the interaction - and . In this case, the speed increments can be represented in this form - and . Therefore, the ratio can be written as -.
Impulse (the amount of energy of a material point) is a vector equal to the product of the mass of a material point and the vector of its velocity —
Impulse of the system (amount of motion of the system of material points) is the vector sum of the impulses of material points that this system consists of - .
It can be concluded that in the case of a closed system, the momentum before and after the interaction of material points must remain the same - , where and . It is possible to formulate the law of conservation of momentum.
The momentum of an isolated system remains constant in time, regardless of the interaction between them.
Required definition:
Conservative forces - forces, the work of which does not depend on the trajectory, but is due only to the initial and final coordinates of the point.
Formulation of the law of conservation of energy:
In a system in which only conservative forces act, the total energy of the system remains unchanged. Only transformations of potential energy into kinetic energy and vice versa are possible.
The potential energy of a material point is a function of only the coordinates of this point. Those. potential energy depends on the position of the point in the system. Thus, the forces acting on a point can be defined as follows: can be defined as: . is the potential energy of a material point. 
Multiply both sides by and we get 
. We transform and obtain an expression proving law of energy conservation
. 
Elastic and Inelastic Collisions
Absolutely inelastic impact - a collision of two bodies, as a result of which they are connected and then move as one.
Two balls , s and experience a perfectly inelastic gift with each other. According to the law of conservation of momentum. From here we can express the speed of two balls moving as a whole after the collision - 
. Kinetic energies before and after impact: 
And 
. Let's find the difference
,
where - reduced mass of balls . This shows that in the case of an absolutely inelastic collision of two balls, the kinetic energy of the macroscopic motion is lost. This loss is equal to half the product of the reduced mass times the square of the relative velocity.
4.1. Balls m 1 and m 2 move towards each other with velocities V 1 and V 2 and collide inelastically. Determine the speed of the balls after impact.
4.2. A body of mass 0.5 kg is thrown upwards with a speed of 4 m/s. Determine the work of gravity, kinetic, potential, total energy when lifting the body to maximum height
4.3. A bullet of mass 20 g, flying horizontally at a speed of 200 m/s, hits a bar suspended on a long cord and gets stuck in it. The mass of the bar is 5 kg. Determine the height of the bar after the impact, if before the impact the bar was moving at a speed of 0.1 m / s towards the bullet.
4.4. A man stands on a stationary trolley and throws a load of 8 kg horizontally at a speed of 10 m/s. Determine the work done by him at the moment of throwing, if the mass of the cart together with the person is 80 kg. At what distance from a stone that fell to the Earth 0.5 s after the throw will the cart stop? if the coefficient of friction is 0.1.
4.5. There is a 60 kg fisherman in a 240 kg boat. The boat is moving at a speed of 2m/s. A person jumps from a boat in a horizontal direction at a speed of 4m/s relative to the boat. Find the speed of the boat after the jump of the person in the direction opposite to the movement of the boat.
4.6. An anti-aircraft projectile explodes at the top of the trajectory into three fragments. The first and second fragments scattered at right angles to each other, and the speed of the first fragment with a mass of 9.4 kg is 60 m/s and is directed in the same direction, and the speed of the second fragment with a mass of 18 kg is 40 m/s. The third fragment flew up at a speed of 200m/s. Determine the mass and speed of the projectile before breaking.
4.7. In a closed system, a body in which only elastic forces and gravity. The change in potential energy is 50J. What is the work done by the forces acting in this system? Determine the change in kinetic energy, total mechanical energy systems.
4.8. On a railway platform with a mass of 16 tons, a gun with a mass of 4 tons is installed, the barrel of which is directed at an angle of 60 degrees to the horizon. With what speed did a projectile with a mass of 50 kg fly out of the gun if the platform stopped after the shot, having traveled a distance of 3 m in 6 s?
4.9. A body is thrown upwards at an angle to the horizon with a speed V 0 . Determine the speed of this body at height h above the horizon. Does the modulus of this speed depend on the angle of throw? Air resistance is ignored.
4.10. A skater, standing on ice, throws a load of 5 kg horizontally at a speed of 10 m/s. At what distance will the skater roll back if his mass is 65 kg, the coefficient of friction is 0.04.
4.11. The boat is stationary standing water. A person, moving evenly, moves from the bow of the boat to the stern. At what distance will the boat move if the masses of a person and a boat are respectively 60 kg and 120 kg, and the length of the boat is 3 m?
4.12. What is the minimum speed the body must have at the bottom point of the "dead loop" with a radius of 8 m, so as not to break away from it at the top point?
4.13. A weight of mass 5g is hanging on a string. The thread is deflected 30 degrees from the vertical and released. What is the tension in the string when the load passes through the equilibrium position?
4.14. The head of a pile hammer with a mass of 0.6 tons falls on a pile with a mass of 150 kg. Find the efficiency of the striker, assuming the impact is inelastic.
4.16. The first body begins to slide without friction along an inclined plane of height h and length nh At the same time, the second body falls from height h. Compare the final velocities of bodies and the time of their movement to the Earth, if air resistance is not taken into account.
4.17. A body of mass 2kg moves towards a second body of mass 1.5kg and collides inelastically with it. The velocities of the bodies before the collision are respectively equal to: 1m/s and 2m/s. How long will the bodies move after the collision if the coefficient of friction is 0.05?
4.18. A circus gymnast kicks from a height of 1.5 m onto a tightly stretched net. What will be the maximum sag of the gymnast in the net. If, in the case of a calmly lying gymnast, the sagging of the net is 0.1 m?
4.19. A man of mass M jumps at an angle to the horizon: α with a speed V 0 . At the top of the trajectory, he throws a stone m with a speed V 1 . How high did the person jump?
4.20. A body slides from the top of a sphere with a radius of 0.3 m. Find Ө,
corresponding to the point of separation of the body from the sphere and the velocity
Bodies at the time of separation.
STATICS. HYDROSTATICS.
B C 5.1. A load of 4 kg is suspended on cords. AD=100cm, SD=SV=
200cm What are the elastic forces of the AD and SD cords?
5.2. A load of mass 400 kg is placed on an inclined plane 5 m long and 3 m high. What force 1) in parallel; 2) perpendicular to the plane must be applied so that the load is kept at rest, the coefficient of friction is 0.2.
5.3. A beam 10 m long rests on two supports with its ends. At a distance of 2m from the edge of the beam lies a load of mass 5t. Determine the vertical reaction forces of the supports if the mass of the beam is 10 tons.
5.4. A pipe with a mass of 2100 tons and a length of 16 m rests on supports located at a distance of 4 m and 2 m from its ends. What is the smallest force that must be applied to lift the pipe: a) by the left edge; b) over the right edge?
5.5. A worker lifts a homogeneous board of mass 40 kg from the Earth at one end so that the board forms an angle of 30 degrees with the horizon. What force is applied perpendicular to the board by the worker, holding the board in this position?
5.6. The top end of the ladder rests on a smooth vertical wall, while the bottom end rests on the floor. The coefficient of friction is 0.5. At what angle of inclination to the horizon will the ladder be in equilibrium?
5.7. A homogeneous rod with a mass of 5 kg rests on a smooth vertical wall and a rough floor, forming an angle of 60 degrees with it. To move this rod, a horizontal force of 20N was required. Determine the coefficient of friction.
to problem 5.7. to problem 5.8.
5.8. The lower end of the rod AB is hinged. A rope AC is tied to the upper end of A, keeping the rod in balance. Determine the tensile force of the rope if the gravity of the rod is R. It is known that the angle ABC is equal to the angle BCA. Angle CAB is 90 degrees.
5.9. Homogeneous halves of a rod 30 cm long are made of iron, the other of aluminum. The cross-sectional areas of both halves are the same. Where is the center of gravity of the rod?
5.10. At what depth is the submarine if water presses on the roof of the exit hatch with an area of 3 10 3 cm 2 with a force of 1.2 10 6 N?
5.11. The lower base of the hollow cylinder is covered with a light plate and immersed in water to a depth of 37 cm. With what force does the water press the plate if its area is 100 cm 2. What is the minimum height of a column of oil that must be poured into the cylinder so that the plate falls off?
5.12. Mercury is poured into the communicating vessels, and then a column of the investigated liquid 15 cm high is poured into the right knee over the mercury. The upper level of mercury in the left knee is 1 cm higher than in the right. Determine the density of the investigated liquid.
5.13. Mercury is poured into a U-shaped tube, and on top of it, water is poured into one knee, and oil into the other. The mercury levels in both knees are the same. Find the height of the water column if the height of the oil column is 20 cm.
5.14. What is the tensile force of the rope with a uniform rise from the water of a lead casting with a volume of 2 dm 3?
5.15. On one side of the scale lies a piece of silver weighing 10.5 kg, and on the other a piece of glass weighing 13 kg. Which cup will outweigh when the scale is immersed in water?
5.16. A hollow zinc ball with an outer volume of 200 cm3 floats in water. Half loaded. Find the volume of the cavity.
5.17. The weight of a piece of marble in kerosene is 3.8N. Determine its weight in air. Ignore the buoyancy force of the air.
5.18. the small piston of the hydraulic press in one stroke descends by 0.2 m. and the large piston rises by 0.01 m. With what force F 2 acts on the body clamped in it, if the force F 1 \u003d 500N acts on the small piston?
5.19. A hydraulic lift lifts a car weighing 2·10 3 kg. How many strokes does the small piston make in 1 minute, if in one stroke it drops 25 cm? Lift motor power 250W, efficiency-25% Piston area 100 cm 2 and 2 10 3 cm 2
5.20. Fluid flows through a horizontally located pipe of variable cross section. Compare the values of the velocities and pressures of the liquid on the walls of the vessel in sections S 1, S 2, S 3.
6.1. What process happened to the gas? Which equation
R Is this process described? Compare temperatures
1 2 At this transition, the mass does not change.
6.2. Compare volumes in this process. Justify the answer. P 1 Mass does not change
6.3. How did the pressure and density of the gas change?
V 1 Justify your answer. The mass does not change.
6.4. How and how many times will the gas temperature change during the transition
P from state 1 to state 2. P 1 =2P 2 ; V 2 \u003d 3V 1.
6.5. Initial State Options ideal gas R 1, V 1, T 1. The gas is isochorically cooled to T 2 = 0.5T 1, then isothermally compressed to the initial pressure. Draw a graph of this transition in P-T coordinates. Write an equation for each process.
6.6. Indicate the processes that the gas goes through sequentially
during this transition. burn gas laws for each
4 transitions. Draw a graph of this transition in P-V coordinates.
P Indicate the processes that the gas goes through sequentially
4 at this transition.
3 2 Write down the gas laws for each transition.
0 1 T Draw a graph of this transition in P-V, V-T coordinates.
6.8. How many oxygen molecules are contained in a 1 cm 3 flask under normal conditions?
6.9. At 27 degrees Celsius and a pressure of 10 5 Pa, there are 2.45 10 27 air molecules in the room. Calculate the volume of the room.
6.10. A sphere with a diameter of 20 cm contains 7 g of air. To what T can this ball be heated if the maximum pressure that the walls of the ball can withstand is 0.3 MPa?
6.11. Air in a vessel with a volume of 5 liters is at a temperature of 27 degrees Celsius under a pressure of 2 MPa. What mass of air was released from the vessel if the pressure in it dropped to 1 MPa and the temperature dropped to 17 degrees Celsius?
6.12. A 10-liter cylinder contains helium at a pressure of 10 6 Pa at a temperature of 37 degrees Celsius. After 10 g of helium was taken from the balloon, the temperature dropped to 27 degrees Celsius. Determine the pressure of helium remaining in the cylinder.
6.13. In vessels with volumes of 5 l and 7 l there is air at a pressure of 2 10 5 Pa and 10 5 Pa. The temperature in both vessels is the same. What pressure will be established if the vessels are connected to each other. The temperature does not change.
6.14. An ideal gas is under pressure of 2·10 5 Pa at 27 degrees Celsius. Due to the isobaric expansion, V of the gas increased by a factor of 3. Next, the gas is isothermally compressed to the initial V. Determine the final pressure and temperature of the gas. Draw a graph of this process in the coordinates P-V, P-T.
6.15. Nitrogen weighing 7 g is under a pressure of 0.1 MPa and a temperature of 290 K. Due to isobaric heating, nitrogen occupied a volume of 10 liters. Determine the volume of gas before expansion and T of gas after expansion, the density of the gas before and after expansion.
6.16. The cylinder contains a certain amount of gas at a pressure of 1 atm. With the valve open, the cylinder was heated, after which the valve was closed and the gas cooled to 10 degrees Celsius, and the pressure in the cylinder dropped to 0.7 atm. By how many degrees did the cylinder cool down?
6.17. A cylinder with a base area of 250 cm 2 contains 1 g of nitrogen compressed by a weightless piston, on which lies a weight of 5 kg. How much will V of the gas increase? Atmospheric pressure is 1 atm.
6.18. In a glass tube sealed at one end, the length of which is 65 cm. there is a column of air, squeezed from above by a column of mercury 25 cm high, reaching the upper unsealed edge of the tube. The tube is slowly turned over, and part of the mercury is poured out. Atmospheric pressure 75mm Hg What is the height of the mercury column that remains in the tube?
6.19. A cylindrical tube of length L, sealed at one end, is immersed in water until its sealed end remains flush with the surface of the water. When the air and water temperatures in the tube became equal, it turned out that the water in the tube rose by 2/3 L. Determine the initial temperature of the air in the tube if the water temperature is T, and Atmosphere pressure R 0 .
6.20. Define average speed gas molecules, the density of which at a pressure of 9.86 10 4 Pa is 8.2 10 2 kg / m 3. What will be the gas if the values of pressure and density are given for 17 degrees Celsius.
THERMODYNAMICS.
7.1. A monatomic ideal gas goes from state 1 to state 2.
P Find the work done by the gas during the transition, change
0 2 internal energy and the amount of heat imparted to the gas.
0 V R 1 \u003d 10 5 Pa, R 2 \u003d 2 10 5 Pa, V 1 \u003d 1l, V 2 \u003d 2l,
7.2. An ideal monatomic gas in its original state had parameters P 1 =10 5 Pa and V 1 =1m 3 . Then the gas was expanded isobarically to V 2 =5m 3 . Find the work done by the gas during the transition, the change in internal energy and the amount of heat imparted to the gas.
7.3. R 1 \u003d 10 5 Pa, R 2 \u003d R 3 \u003d 3 10 5 Pa, V 1 \u003d V 2 \u003d 1l,
P 2 3 V 3 \u003d 3l.
Find the work done by the gas during the transition, the amount
heat absorbed by the gas per cycle; the amount of heat given off by the gas per cycle; efficiency.
7.4. In the cylinder under the piston there is air Р 1 =10 5 Pa, V 1 =10l. Further, its state changes along a closed loop:
1. V=const, P increases by 2 times; 2. P=const, V is doubled.
3.T=const, V increases by 2 times; 4.P = const, the air returns to its original state.
Draw a graph of this process in P-V coordinates. Indicate in which processes the air absorbs heat, and in which it gives off. Determine from the graph what is equal to useful work per cycle. Consider air as an ideal gas.
7.5. An ideal monatomic gas in an amount of 1 mol completes a closed cycle consisting of two isochores and two isobars. The temperatures at point 1 and 3 are equal.
T 1 \u003d 400K, T 2 \u003d T 1, T 3 \u003d 900K
P 2 3 Indicate in which processes the air absorbs heat, and in which it gives off
Find the work done by the gas per cycle.
7.6. Helium weighing 400g is isochorically heated from 200K to 400K, and then isobarically heated to 600K. Draw a graph of this process in P-V coordinates. Find the work done by the gas during the transition, the change in internal energy and the amount of heat imparted to the gas.
7.7. R 1 \u003d 4 10 5 Pa, R 2 \u003d 10 5 Pa, V 1 \u003d 1l, V 2 \u003d 2l.
P Find the work done by the gas during the transition,
1 change in internal energy and the amount of heat,
2 received by gas.
7.8. 1-2: adiabatic expansion;
2-3: isothermal compression;
T 3-1: isochoric heating.
What is the work done by the gas in the adiabatic process?
1 If during isochoric heating the gas was told
3 2 heat Q 3-1 \u003d 10kJ? What is the cycle efficiency?
V if the gas during isothermal compression gave off heat Q 2-3 \u003d 8kJ?
7.9. Draw a graph of this process in P-V coordinates.
V Indicate in which processes air absorbs heat, and in
which gives.
T Find the work done by the gas during the transition if
R 2 = 4 10 5 Pa, R 1 = R 3 = 10 5 Pa, V 1 = V 2 = 1l V 3 = 4l.
7.10. The mass of an ideal gas - helium is equal to 40g at T=300K is cooled at V=const so that P decreases by 3 times. Then the gas expands at P = const so that its T becomes equal to the original. Find the work done by the gas during the transition, the change in internal energy and the amount of heat imparted to the gas.
7.11. During isobaric heating of some ideal gas in an amount of 2 mol per 90 K, 2.1 kJ of heat was imparted to it. . Find the work done by the gas during the transition, the change in internal energy.
7.12. An ideal monatomic gas with a volume of 1 liter is under a pressure of 1 MPa. Determine how much heat must be imparted to the gas in order to:
1) V increase by 2 times as a result of the isobaric process;
2) Increase P by 2 times as a result of the isochoric process.
7.13. The work of expansion of some monatomic gas is 2 kJ. Determine how much heat it is necessary to inform the gas of a change in internal energy if the process proceeded: isobarically, adiabatically.
7.14. An ideal monatomic gas was given an amount of heat of 20 kJ. Find the work of the gas and the change in internal energy if heating occurred: isobaric, isochoric, isothermal.
7.15. An ideal monatomic gas went through a Carnot cycle. The gas received 5.5 kJ of heat from the heater and did 1.1 kJ of work. Determine the efficiency, T 1 /T 2.
7.16. An ideal monatomic gas has completed the Cornot cycle. 70% of the amount of heat received from the heater is given to the refrigerator. The amount of heat received from the heater is 5kJ. Determine the efficiency of the cycle, the work done during the full cycle.
7.17. There is an ideal monatomic gas with a volume of 0.01 m 3 at a pressure of 0.1 MPa and a temperature of 300K. The gas was heated at V=const to 320K, and then heated at P=const to 350K. Find the work done by the gas during the transition, the change in internal energy and the amount of heat absorbed by the gas during the transition from state 1 to state 3. Draw a graph of this process in P-V coordinates.
7.18. In a cylinder with a volume of 190 cm 3, under the piston, there is a gas having a temperature of 323K. Determine the work of expansion of the gas when it is heated by 100K, if the weight of the piston is 1200N, the area is 50 cm 3 and the atmospheric pressure is 100kPa.
7.19. A cycle is completed with 3 moles of an ideal monatomic gas.
R 2 3 Gas temperature in various states: 1- 400K; 2- 800K;
1 4 3- 2400K; 4- 1200K. Determine the gas work per cycle and efficiency
T cycle. Draw a graph of this process in P-V coordinates. 7.20. Initially, 1 mole of a monatomic gas was in an insulated vessel with a movable lid, occupying V 1 , at a pressure of P 1 and a temperature of 27 degrees Celsius. Then it was heated with a heater, which imparted to the gas an amount of heat of 30 kJ. As a result, the gas expanded at P=const, heating up to T 2 and occupying V 2 . Determine the work of the gas during expansion, T 2, V 1 / V 2.
HEAT.
8.1. A piece of ice was put into a vessel containing 10 kg of water at a temperature of 10 degrees Celsius at a temperature of -50 degrees Celsius, after which the temperature of the resulting ice mass turned out to be -4 degrees Celsius. What amount of ice m 2 was put into the vessel? Draw a heat transfer diagram in t-Ө coordinates.
8.2. A bath with a capacity of 100 liters must be filled with water having Ө=30 degrees Celsius, using water at 80 degrees Celsius and ice at a temperature of -20 degrees Celsius. Determine the mass of ice to be placed in the bath. Ignore the heat capacity of the bath and the heat loss. Draw a heat transfer diagram in t-Ө coordinates.
8.3. A heat-insulated vessel contains a mixture of water weighing 500 g and ice weighing 50 g at a temperature of 0 degrees Celsius. Dry saturated steam weighing 50 g at a temperature of 100 degrees Celsius. What will be the temperature of the mixture after thermal equilibrium is established? Draw a heat transfer diagram in t-Ө coordinates.
8.4. A mixture consisting of 5 kg of ice and 15 kg of water at a total temperature of 0 degrees Celsius must be heated to Ө=80 degrees Celsius by passing water vapor at a temperature of 100 degrees Celsius. Determine the required amount of steam. Draw a heat transfer diagram in t-Ө coordinates.
8.5. To what temperature must an aluminum cube be heated so that, when placed on ice, it is completely immersed in it?
8.6. An iron calorimeter weighing 0.1 kg contains 0.5 kg of water at a temperature of 15 degrees Celsius. Lead and aluminum with a total mass of 0.15 kg are thrown into the calorimeter at 100 degrees Celsius. As a result, the water temperature rose to Ө=17 degrees Celsius. Determine the masses of lead and aluminum.
8.7. A calorimeter containing 250 g of water at 15 degrees Celsius is thrown into 20 g of wet snow. The temperature in the calorimeter dropped to Ө= 10 degrees Celsius. How much water was in the snow?
8.8. With what speed does a meteorite flies into the Earth's atmosphere, if at the same time it heats up, melts and turns into steam? Meteoric matter is made up of iron. The initial temperature of the meteor is 273 degrees Kelvin.
8.9. How much coal m 2 is required to spread m 1 = 1 ton of gray cast iron taken at 50 degrees Celsius? Cupola efficiency 60%.
8.10. A lead weight falls to the ground and hits an obstacle. The speed of the kettlebell at impact is 330m/s. Calculate how much of the weight will melt if all the heat released during the impact is absorbed by the weight. The temperature of the kettlebell before impact is 27 degrees Celsius.
8.1. Two identical pieces of ice fly towards each other at the same speed and turn into steam on impact. Rate minimum possible speeds ice floes before impact if their initial temperature is -12 degrees Celsius.
8.12. From what height must a tin ball fall so that when it hits the Earth it is completely glorified. Assume that 95% of the energy of the ball was spent on heating and melting it. The initial temperature of the ball is 20 degrees Celsius.
8.13. In a snow melter, the efficiency of which is 25%, 2 tons of dry firewood were burned. What area can be freed from snow at -5 degrees Celsius when burning this amount of fuel, if the snow is 50 cm thick.
8.14. How much snow at 0 degrees Celsius will melt under the wheels of a Volga car if it skids for 10 seconds? 1% of all its power is used for slipping. The power of the car is 55.2 kW.
8.15. The car traveled a distance of 120 km at a speed of 72 km/h. 19kg of gasoline was used on this route. What is the average power developed by the car during the run, if the efficiency is 75%?
8.16. In electric stoves with an efficiency of 84%, a 2-liter kettle is heated from 10 degrees Celsius to 100 degrees Celsius, and m 2 \u003d 0.1 m part of the water boils away. The heat capacity of the teapot is 210J/K. What is the power of the tile if the heating of water lasted 40 minutes?
8.17. How long does it take to heat a mass of 2 kg of ice taken at -16 degrees Celsius on an electric stove with a power of 600 W at an efficiency of 75% in order to turn it into water, and heat the water to 100 degrees Celsius?
8.18. In the manufacture of shot, molten lead is poured into water in drops at a solidification temperature. How much lead was poured into water weighing 5 kg if its temperature rose from 15 degrees Celsius to Ө=25 degrees Celsius.
8.19. Find the amount of heat released during an absolutely inelastic collision of two balls moving towards each other. The mass of the first ball is 0.4 kg, its speed is 3 m/s, the mass of the second is 0.2 kg, the speed is 12 m/s.
8.20. In a copper vessel heated to 350 degrees Celsius, put m 2 = 600 g of ice at a temperature of -10 degrees Celsius. As a result, m 3 \u003d 550 g of ice mixed with water appeared in the vessel. Find the mass of the vessel.
ELECTROSTATICS.
9.1. Two identically charged balls of mass 0.5 g, suspended at one point on threads 1 m long, parted so that the angle between them became right. Determine the charges of the balls.
9.2. Two identical charged balls, located at a distance of 0.2 m, attracted with a force of 4 10 -3 N. After the balls were brought into contact and then separated to the same distance, they began to repel with a force of 2.25 10 -3 N Determine the initial charges of the balls.
9.3. Charges 10 -9 C, - 10 -9 C and 6 10 -9 C are located at the corners of a regular triangle with a side of 20cm. What is the direction of the force acting on the third charge. What is it equal to?
9.4. Three identical charges of 10 -9 C are located at the vertices of a triangle with legs 10cm and 30cm. Define tension electric field created by all charges at the point of intersection of the hypotenuse with the perpendicular dropped on it from the vertex of the right angle.
9.5. At the vertices of the square there are charges 1/3 10 -9 C, -2/3 10 -9 C, 10 -9 C,
4/3 10 -9 Cl. Determine the potential and strength of the electric field in the center of the square. The diagonal of the square is 2a=20cm.
9.6. Determine the potential and strength of the electric field at points B and C, located from a charge of 1.67 10 -7 C at distances of 5 cm and 20 cm. Define a job electrical forces when moving charge q 0 \u003d 10 -9 C from point B to point C.
9.7. A copper ball with a radius of 0.5 cm is placed in oil with a density of 0.8·10 3 kg/m 3 . Determine the charge of the ball if the ball is suspended in oil in a uniform electric field. The electric field is directed upwards and its strength is 3.6·10 5 V/m.
9.8. Two point charges: 7.5 nC and -14.7 nC are located at a distance of 5 cm. Determine the electric field strength at a point located at a distance of 3 cm from a positive charge and 4 cm from a negative charge.
9.9. Two point charges: 3·10 -8 C and 1.33K·l10 -8 C are located at a distance of 10 cm. Find a point on the straight line connecting these charges, the electric field strength at which is equal to 0. What is the potential of the electric field at this point?
9.10. Two point charges: 1nC and 3nC are located at a distance of 10cm. At what points of the electric field on the straight line connecting these charges, the electric field strength is 0? Solve the problem for two cases: 1) charges of the same name; 2) charges have different signs. Calculate the potential of points where the field strength is 0.
9.11. The field is created by a point charge 2·10 -6 C. When moving q 0 =-5·10 -7 C in this field from point 1 to point 2, an energy of 3.75·10 -3 J is released. The potential of the point is 1:1500V. What is the potential of point 2? What is the distance between the points?
Q 1 Q 2 VA What work needs to be done in order to move q 0 \u003d -5 10 -8 C from point A to point B in the field of two point charges 3nC and -3nC. The distance between the charges is 10cm, the distance from the second charge to point B is 20cm, the distance from point B to point A is 10cm.
9.13. Two point charges: 6.6 10 -9 C, 1.32 10 -6 C are located at a distance of 10 cm. What work must be done to bring them closer to a distance of 25 cm?
9.14. How many electrons does a charged dust grain with a mass of 10 -11 g contain if it is in equilibrium between two horizontal parallel plates charged to a potential difference of 16.5 V? The distance between the plates is 5mm. With what acceleration and in what direction will a dust particle move if it loses 20 electrons?
9.15. An electron flies out of point A, the potential of which is 600V at a speed of 12 10 6 m/s in the direction lines of force fields. At what distance from point A will the electron stop? Determine the potential of point B of the electric field, reaching which after 10 -6 s the electron will stop.
9.16. A charge is placed on a ball with a radius of 2 cm: 6.4 10 -12 C. With what speed does an electron fly up to it, starting from a point infinitely distant from the ball?
9.17. An electron flies into a flat capacitor with a speed of 2·10 7 m/s, directed parallel to the capacitor plates. Write down the equation of electron motion along the x-axis, parallel to the plates, and along the Y-axis, perpendicular to the x-axis. At what distance y 1 from its original direction will the electron move during the flight in the capacitor, if the distance between the plates is 2 cm, the length of the capacitor plates is 5 cm. Potential difference between plates 200V?
9.18. q 1 C Two point charges: 2 10 -6 C, 15 10 -6 C are located at a distance
L + q 0 40 cm at points A and B. Along the SD parallel to AB, at a distance of 30 cm from
her, the charge q 0 =10 -8 C moves slowly. Define a job
q 2 D electric forces when moving a charge from point C to point D.
9.19. The distance between the plates of a flat capacitor is 4 cm. The electron starts moving from the “-” charged plate at the moment when the proton starts moving from the “+” plate. Write down the equations of motion inside the capacitor for the electron and proton. At what distance from the “+” plate will the electron and proton meet?
9.20. An electron flies into a flat capacitor 5 cm long at an angle of 15 degrees to the plates. An electron has an energy of 1500 eV. The distance between the plates is 1 cm. Determine the potential difference across the plates of the capacitor, at which the electron, when leaving the capacitor, will move parallel to the plates.
ELECTRICAL CAPACITY.
10.1. The charge of the first ball is 2 10 -7 K, the second is 10 -7 C. The capacitance of the balls is 2pF and 3pF. Determine the charges of the balls after they are connected with a wire.
10.2. A ball with a diameter of 20 cm is charged with a charge of 333 10 -9 C. What charge must be added to this ball in order for its potential to increase by 6000V? What will be the potential of the ball?
10.3. On one ball with a diameter of 8cm there is a charge of 7·10 -9 C, and on the other ball with a diameter of 12cm there is a charge of 2·10 -9 C. These balls are connected with wire. Will the charge move and in what direction, and in what quantity?
10.4. A charged ball with a radius of 20 cm, having a potential of 1000 V, is connected to an uncharged ball with a long wire. After connecting the balls, their potential is 300V Determine the radius of the second ball.
10.5. A capacitor with a capacity of C 0 charged to a certain potential difference was connected in parallel to the same uncharged capacitor. How will the charge, electric field strength, potential difference, energy change in the first capacitor?
10.6. A flat air capacitor C 0 was charged from a source to a certain potential difference and it has a charge q 0. After disconnecting from the source, the distance between the plates was reduced by 2 times. How will the capacitance, charge, potential difference, energy change when the capacitor plates approach each other?
10.7. In a flat charged capacitor, disconnected from the current source, an ebonite plate with a dielectric constant of 3 was replaced with a porcelain one with a dielectric constant of 6. The plates fit snugly against the capacitor plates. How will the capacitance, charge, potential difference, energy of a flat capacitor change?
10.8. A square flat capacitor with a side of 10 cm was given a charge of 10 -9 C.
The distance between the plates is 5mm. What is the capacitance of the capacitor, the tension inside the capacitor? What force acts on a test charge of 10 -9 C, located between the capacitor plates? How does this force depend on the location of the test charge?
10.9. If you charged yourself to a potential of 15V by dragging your feet across the floor, how much energy would you store? You are a sphere with a radius of 50 cm and a surface area approximately equal to the surface of your body.
10.10. What charge will pass through the wires connecting the flat capacitor plates to the battery terminals when the capacitor is immersed in kerosene? The area of the capacitor plates is 150 cm 2, the distance between the plates is 5 mm, the battery EMF is 9.42, with a dielectric constant equal to 2.
10.11. A flat air capacitor was charged to a potential difference of 200V, then disconnected from the source. What will be the potential difference between the capacitor plates if the distance between them is increased from the initial 0.2 mm to 7 mm, and the space between the plates is filled with mica with a dielectric constant equal to 7?
10.12. A 20µF capacitor charged to a potential difference of 100V was connected in parallel with a capacitor charged to a potential difference of 40V, the capacitance of which is unknown. Determine the capacitance of the second capacitor if the potential difference on the capacitor plates after the connection is 80V (the plates were connected with the same charges).
10.13. A capacitor charged to a potential difference of 20V was connected in parallel with another capacitor charged to a potential difference of 4V, the capacitance of which was 33 μF. Determine C 1 if the potential difference on the capacitor plates after the connection is 2V (the plates were connected by opposite charges).
10.14. A 4µF capacitor was charged to a potential difference of 10V. What charge will be on the capacitor plates if it is connected in parallel with another capacitor, the capacitance of which is 6 μF, charged to a potential difference of 20 V? Capacitor plates with opposite charges are connected.
10.15. Two identical flat air capacitors with a capacity of 1 μF are connected in parallel and charged to a potential difference of 6V. What will be the potential difference between the plates of the capacitor, if after disconnecting the capacitors from the source at one capacitor, the distance between the plates of 5 mm is reduced by 2 times. What is the capacitance of the capacitor bank, the field strength between the plates of the first and second capacitors after reducing the distance?
10.16. A battery of three series-connected capacitors with capacities: 100pF, 200pF, 500pF is connected to the battery, which reported the charge of 33·10 -9 C to the battery. Determine the potential difference across each capacitor, the emf of the battery, the total capacitance of the capacitor bank
10.17. Between the plates of a charged capacitor, a dielectric plate with a dielectric constant equal to 6 is pushed in tightly. Compare the charges of the capacitors, the potential difference on the plates, the capacitance of the condensates, the intensity, and the energy before and after the introduction of the dielectric plate. Consider the cases: 1) the capacitor is disconnected from the source; 2) the capacitor is connected to the source.
10.18. The area of the plates of a flat air capacitor is 0.01m 2, the potential difference is 280V, the charge of the plates is a charge of 495 10 -9 C. Determine the field strength inside the capacitor, the distance between the plates, the speed that the electron received. Having passed the path from one plate to another in the capacitor, the energy of the capacitor, the energy density, the capacitance of the capacitor.
10.19. The area of the plates of a flat air capacitor is 0.01m 2, the distance between the plates is 1mm. A potential spread of 0.1 kV was applied to the capacitor plates. The plates are moved apart to a distance of 25 mm. Determine the field strength inside the capacitors, capacitance, energy before and after the expansion of the plates, if the voltage source before the separation: 1) was not turned off; 2) turned off.
10.20. A flat capacitor is filled with a dielectric and a certain potential difference is applied to its plates. Its energy is 20 μJ. After disconnecting the capacitor from the voltage source, the dielectric was removed from it. The work of external forces against the forces of the electric field when the dielectric is removed is 700 μJ. Find the permittivity.
D.C.
11.1 The voltmeter is designed to measure the maximum voltage of 3V. The resistance of the device is 300 ohms. The number of scale divisions of the device is 100. What will be the value of the scale division of the device if you use it as a milliammeter?
11.2. Find the resistance of a copper wire with a mass of 1 kg and an area of 0.1 mm 2.
11.3. When included in the electrical circuit of a conductor with a diameter of 0.5 mm and a length of 47 cm, the voltage is 12V, the current strength is 1A. Find the resistivity of the conductor.
11.4. Electrical circuit consists of three series-connected pieces of wire of the same length, made of the same material, but having different sections: 1mm, 2mm 3mm. The voltage at the ends of the circuit is 11V. Determine the voltage on each conductor.
11.5. The ammeter shows 0.04 A, and the voltmeter 20V. Determine the resistance of the voltmeter if the resistance of the conductor is 1 kΩ.
11.6. In the current source circuit with an EMF of 30V, a current of 3A flows. The voltage at the source terminals is 18V. Determine the external resistance of the circuit and the internal resistance of the source.
11.7. In a circuit consisting of a rheostat and a source with an EMF of 6V and an internal resistance of 2Ω, a current of 0.5A flows. What current will pass when the resistance of the rheostat decreases by 3 times?
11.8. Two conductors of the same material, having the same length and different cross section (the cross section of the first is 2 times larger than the second) are connected in series. Compare conductor resistances. The amount of heat released in these conductors during the passage of current in them and the change in their temperature. Assume that all the heat released is used to heat the conductors.
11.9. The lamp is connected by copper wires to a source with an EMF of 2V and an internal source resistance of 0.04 Ohm, the length of the wires is 4 m, their diameter is 0.8 mm. The voltage at the source terminals is 1.98V. Find the resistance of the lamp.
