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1. Mechanical motion Kinematics of a rigid body
The line along which the point of the body moves is called the trajectory of motion. Mechanical motion is the process of changing the position of a body in space relative to other bodies over time. 2 1 ℓ s
2. Relativity of mechanical motion. Reference systems.
Mechanical motion is relative, the expression "the body moves" is meaningless until it is determined in relation to what the motion is considered. To determine the position of a material point at any time, select: Reference body Coordinate system Clock Reference body is a body relative to which the position of other (moving) bodies is determined.
Coordinate systems Coordinate line Examples: elevator, metro tram. Chess coordinate plane, Spatial coordinate system x A (x) x y A (x, y) x y z A (x, y, z) treasure, chandelier,
Mechanical motion is characterized by three physical quantities: displacement, speed and acceleration. A directed straight line segment drawn from the initial position of a moving point to its final position is called displacement (). Displacement is a vector quantity. The unit of movement is the meter. 3. Characteristics of the mechanical movement
Speed is a vector physical quantity that characterizes the speed of movement of a body, numerically equal to the ratio of movement in a small period of time to the value of this gap. The time interval is considered sufficiently small if the speed during uneven movement during this interval did not change. The formula for instantaneous velocity has the form. The SI unit of speed is m/s. In practice, the speed unit used is km/h (36 km/h = 10 m/s). Measure speed with a speedometer.
Acceleration is measured with an accelerometer. If the speed changes the same throughout the entire time of movement, then acceleration can be calculated by the formula: Unit of acceleration - Acceleration - a vector physical quantity that characterizes the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.
The characteristics of mechanical motion are interconnected by the main kinematic equations: If the body moves without acceleration, then its speed does not change for a long time, a \u003d 0, then the kinematic equations will look like:
four . Types of movement and their graphic description.
Curvilinear Rectilinear By type of trajectory Uneven Uniform By speed Types of movement differ:
If the speed and acceleration of the body have the same direction (a > 0), then such an equally variable motion is called uniformly accelerated. In this case, the kinematic equations look like this:
If the velocity and acceleration of a body are in opposite directions (and
Graphical representation of uniformly variable motion Acceleration versus time
Graphical representation of uniformly variable motion uniformly accelerated uniformly slowed down The module of displacement is numerically equal to the area under the graph of the dependence of the body's speed on time. Speed versus time
Graphical representation of uniformly alternating motion uniformly accelerated uniformly slowed down. Dependence of the coordinate on time along the X axis (x 0 \u003d 0; V 0 \u003d 0)
Connection of the projection of the body displacement with a finite velocity in the case of uniformly accelerated motion. From the equations and you can get: When we get:
5. Fixing 1. Mechanical movement is called ________ 2. The section "Mechanics" consists of _______________ 3. Kinematics studies _________________________ 4. To determine the position of the body, you need to choose ___ 5. Coordinate systems are ___________________ 6. List the physical quantities that characterize the mechanical movement: 7. The line along which the body moves is called __ 8. Displacement is ____________________________ 9. Physical quantity characterizing the rate of change in the speed of the body , is called __________ 10. Write down the equation for the velocity of a body with uniformly accelerated motion of a body with initial speed, different from zero.
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Presentation on the topic: Rotational motion of a rigid body
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The rotational motion of a rigid body or system of bodies is such a motion in which all points move along circles whose centers lie on one straight line, called the axis of rotation, and the planes of the circles are perpendicular to the axis of rotation. The rotational motion of a rigid body or system of bodies is such a motion in which all points move along circles whose centers lie on one straight line, called the axis of rotation, and the planes of the circles are perpendicular to the axis of rotation. The axis of rotation can be located inside the body and outside it, and depending on the choice of the reference system, it can be either moving or stationary. Euler's rotation theorem states that any rotation of three-dimensional space has an axis.
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Kinematics of rotary motion……………………….…….4 Kinematics of rotary motion……………………….…….4 Dynamics of rotary motion………………………………. 13 The basic equation of the dynamics of rotational motion……14 Dynamics of arbitrary motion………………………………..……….26 Conservation laws……………………………………………… ……….....30 Law of conservation of angular momentum……………………………………….31 Kinetic energy of a rotating body……………………………….52 Law of conservation of energy… ……………………….………………………….…57 Conclusion…………………………………………………………………. .…..61 Information materials used ..…………...66
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Example: plane-parallel movement of a wheel without slipping on a horizontal surface. Wheel rolling can be represented as the sum of two movements: translational motion at the speed of the center of mass of the body and rotation about an axis passing through the center of mass. Example: plane-parallel movement of a wheel without slipping on a horizontal surface. Wheel rolling can be represented as the sum of two movements: translational motion at the speed of the center of mass of the body and rotation about an axis passing through the center of mass.
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The kinematics of the movement of the Palace Bridge in St. Petersburg was captured by the method of sequential shooting. Exposure 6 seconds. What information about the movement of the bridge can be extracted from the photo? Analyze the kinematics of its movement. The kinematics of the movement of the Palace Bridge in St. Petersburg was captured by the method of sequential shooting. Exposure 6 seconds. What information about the movement of the bridge can be extracted from the photo? Analyze the kinematics of its movement.
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Kikoin A.K. Kinematic formulas for rotational motion. "Quantum", 1983, No. 11. Kikoin A.K. Kinematic formulas for rotational motion. "Quantum", 1983, No. 11. Fistul M. Kinematics of plane-parallel motion. "Quantum", 1990, No. 9 Chernoutsan A.I. When everything revolves around... "Kvant", 1992, No. 9. Chivilev V., Movement in a circle: uniform and uneven. "Quantum", 1994, No. 6. Chivilev V.I. Kinematics of rotational motion. "Quantum", 1986, No. 11.
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The dynamics of the translational motion of a material point operates with such concepts as force, mass, momentum. The dynamics of the translational motion of a material point operates with such concepts as force, mass, momentum. The acceleration of a translationally moving body depends on the force acting on the body (the sum of the acting forces) and the mass of the body (Newton's second law):
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Design and principle of operation of the device Design and principle of operation of the device Investigation of the dependence of the angular acceleration of disk rotation on the moment of the acting force: on the value of the acting force F at a constant value of the arm of the force relative to the given axis of rotation d (d = const); from the shoulder of the force relative to the given axis of rotation at a constant acting force (F = const); from the sum of the moments of all forces acting on the body about a given axis of rotation. Investigation of the dependence of angular acceleration on the properties of a rotating body: on the mass of a rotating body at a constant moment of forces; on the distribution of mass relative to the axis of rotation at a constant moment of forces. Experimental results:
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The fundamental difference is that mass is invariant and does not depend on how the body is moving. The moment of inertia changes when the position of the axis of rotation or its direction in space changes. The fundamental difference is that mass is invariant and does not depend on how the body is moving. The moment of inertia changes when the position of the axis of rotation or its direction in space changes.
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The theorem on the transfer of axes of inertia (Steiner): the moment of inertia of a rigid body about an arbitrary axis I is equal to the sum of the moment of inertia of this body I0 about the axis passing through the center of mass of the body parallel to the axis under consideration, and the product of the body mass m and the square of the distance d between the axes: transfer of the axes of inertia (Steiner): the moment of inertia of a rigid body about an arbitrary axis I is equal to the sum of the moment of inertia of this body I0 about the axis passing through the center of mass of the body parallel to the considered axis, and the product of the body mass m and the square of the distance d between the axes:
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How do the moments of inertia of the cubes about the axes OO and O'O' differ? How do the moments of inertia of the cubes about the axes OO and O'O' differ? Compare the angular accelerations of the two bodies shown in the figure, with the same action of the moments of external forces on them.
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Task: On a smooth inclined plane A ball and a solid cylinder of the same mass roll down. Which of these bodies Problem: A ball and a solid cylinder of the same mass roll down a smooth inclined plane. Which of these bodies will roll faster? Note: The equation of the dynamics of the rotational motion of the body can be written not only relative to a fixed or uniformly moving axis, but also relative to an axis moving with acceleration, provided that it passes through the center of mass of the body and its direction in space remains unchanged.
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The problem of the rolling of a symmetrical body on an inclined plane. The problem of the rolling of a symmetrical body on an inclined plane. With respect to the axis of rotation passing through the center of mass of the body, the moments of the forces of gravity and the reaction of the support are equal to zero, the moment of the friction force is equal to M = Ftr. Make up a system of equations, applying: the basic equation of the dynamics of rotational motion for a rolling body; Newton's second law for the translational motion of the center of mass.
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The moment of inertia of a ball and a solid cylinder, respectively, are equal The moment of inertia of a ball and a solid cylinder, respectively, are equal Equation of rotational motion: Equation of Newton's second law for the translational motion of the center of mass Acceleration of the ball and cylinder when rolling down an inclined plane, respectively, are equal: ab > ac, therefore, the ball will roll faster than the cylinder. Generalizing the result obtained to the case of rolling of symmetrical bodies from an inclined plane, we find that a body with a smaller moment of inertia will roll faster.
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Arbitrary motion of a rigid body can be decomposed into translational motion, in which all points of the body move at the speed of the center of mass of the body, and rotation around the center of mass. Arbitrary motion of a rigid body can be decomposed into translational motion, in which all points of the body move at the speed of the center of mass of the body, and rotation around the center of mass.
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The sequential shooting mode allows illustrating the theorem on the movement of the center of mass of the system: when the shutter is released, several images can be captured in one second. When such a series is combined, athletes performing tricks and animals in motion turn into a dense line of twins. The sequential shooting mode allows illustrating the theorem on the movement of the center of mass of the system: when the shutter is released, several images can be captured in one second. When such a series is combined, athletes performing tricks and animals in motion turn into a dense line of twins.
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The law of conservation of angular momentum - one of the most important fundamental laws of nature - is a consequence of the isotropy of space (symmetry with respect to rotations in space). The law of conservation of angular momentum - one of the most important fundamental laws of nature - is a consequence of the isotropy of space (symmetry with respect to rotations in space). The law of conservation of angular momentum is not a consequence of Newton's laws. The proposed approach to the conclusion of the law is of a private nature. With a similar algebraic form of writing, the laws of conservation of momentum and angular momentum as applied to one body have a different meaning: in contrast to the speed of translational motion, the angular velocity of rotation of the body can change due to a change in the moment of inertia of the body I by internal forces. The law of conservation of angular momentum is fulfilled for any physical systems and processes, not only mechanical ones.
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The angular momentum of a system of bodies remains unchanged for any interactions within the system, if the resulting moment of external forces acting on it is equal to zero. The angular momentum of a system of bodies remains unchanged for any interactions within the system, if the resulting moment of external forces acting on it is equal to zero. Consequences from the law of conservation of angular momentum in the event of a change in the rotation speed of one part of the system, the other will also change the rotation speed, but in the opposite direction in such a way that the angular momentum of the system does not change; if the moment of inertia of a closed system changes during rotation, then its angular velocity also changes in such a way that the angular momentum of the system remains the same in the case when the sum of the moments of external forces about a certain axis is equal to zero, the angular momentum of the system about the same axis remains constant . Experimental verification. Experiments with Zhukovsky's bench Limits of applicability. The law of conservation of angular momentum is fulfilled in inertial frames of reference.
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The Zhukovsky bench consists of a frame with a support ball bearing in which a round horizontal platform rotates. The Zhukovsky bench consists of a frame with a support ball bearing in which a round horizontal platform rotates. The bench with the person is brought into rotation, inviting him to spread his arms with dumbbells to the sides, and then sharply press them to his chest.
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The law of conservation of angular momentum is fulfilled if: The law of conservation of momentum is fulfilled if: the sum of the moments of external forces is equal to zero (the forces may not be balanced in this case); the body moves in a central force field (in the absence of other external forces; relative to the center of the field) The law of conservation of angular momentum is applied: when the nature of the change over time of the forces of interaction between parts of the system is complex or unknown; about the same axis for all moments of impulse and forces; both fully and partially isolated systems.
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A remarkable feature of rotational motion is the property of rotating bodies in the absence of interactions with other bodies to keep unchanged not only the angular momentum, but also the direction of the axis of rotation in space. A remarkable feature of rotational motion is the property of rotating bodies in the absence of interactions with other bodies to keep unchanged not only the angular momentum, but also the direction of the axis of rotation in space. Daily rotation of the Earth. Gyroscopes Helicopter Circus rides Ballet Figure skating Gymnastics (somersaults) Diving Sports
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The constant reference point for travelers on the surface of the Earth is the North Star in the constellation Ursa Major. The axis of rotation of the Earth is directed approximately to this star, and the apparent immobility of the North Star over the centuries clearly proves that during this time the direction of the axis of rotation of the Earth in space remains unchanged. The constant reference point for travelers on the surface of the Earth is the North Star in the constellation Ursa Major. The axis of rotation of the Earth is directed approximately to this star, and the apparent immobility of the North Star over the centuries clearly proves that during this time the direction of the axis of rotation of the Earth in space remains unchanged.
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A gyroscope is any heavy symmetrical body rotating around the axis of symmetry with a high angular velocity. A gyroscope is any heavy symmetrical body rotating around the axis of symmetry with a high angular velocity. Examples: bicycle wheel; hydroelectric turbine; propeller. Properties of a free gyroscope: keeps the position of the axis of rotation in space; impact resistant; inertialess; has an unusual reaction to the action of an external force: if the force tends to rotate the gyroscope about one axis, then it rotates around the other, perpendicular to it - it precesses. Has a wide range of applications.
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Many features of the behavior of a helicopter in the air are dictated by the gyroscopic effect. A body untwisted along an axis tends to keep the direction of this axis unchanged. Many features of the behavior of a helicopter in the air are dictated by the gyroscopic effect. A body untwisted along an axis tends to keep the direction of this axis unchanged. Turbine shafts, bicycle wheels, and even elementary particles, such as electrons in an atom, have gyroscopic properties.
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The property of the angular velocity of rotation of a body to change due to the action internal forces used by athletes and ballet dancers: when, under the influence of internal forces, a person changes his posture, pressing his hands to the body or spreading them apart, he changes the moment of momentum of his body, while the moment of momentum is preserved both in magnitude and in direction, therefore, the angular velocity of rotation also changes. Athletes and ballet dancers use the property of the angular velocity of rotation of the body to change due to the action of internal forces: when, under the influence of internal forces, a person changes his posture, pressing his arms to the body or spreading them apart, he changes the moment of momentum of his body, while the moment of momentum is preserved as magnitude and direction, so the angular velocity of rotation also changes.
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A skater who rotates around a vertical axis, at the beginning of the rotation, brings his hands closer to the body, thereby reducing the moment of inertia and increasing the angular velocity. At the end of the rotation, the reverse process occurs: when the arms are spread, the moment of inertia increases and the angular velocity decreases, which makes it easy to stop the rotation and proceed to another element. A skater who rotates around a vertical axis, at the beginning of the rotation, brings his hands closer to the body, thereby reducing the moment of inertia and increasing the angular velocity. At the end of the rotation, the reverse process occurs: when the arms are spread, the moment of inertia increases and the angular velocity decreases, which makes it easy to stop the rotation and proceed to another element.
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The gymnast performing somersaults, in the initial phase, bends his knees and presses them to his chest, thereby reducing the moment of inertia and increasing the angular velocity of rotation around the horizontal axis. At the end of the jump, the body straightens, the moment of inertia increases, and the angular velocity decreases. The gymnast performing somersaults, in the initial phase, bends his knees and presses them to his chest, thereby reducing the moment of inertia and increasing the angular velocity of rotation around the horizontal axis. At the end of the jump, the body straightens, the moment of inertia increases, and the angular velocity decreases.
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The push experienced by the jumper into the water, at the moment of separation from the flexible board, “twirls” it, giving the initial stock of angular momentum relative to the center of mass. The push experienced by the jumper into the water, at the moment of separation from the flexible board, “twirls” it, giving the initial stock of angular momentum relative to the center of mass. Before entering the water, having made one or more revolutions with a high angular velocity, the athlete extends his arms, thereby increasing his moment of inertia and, consequently, reducing his angular velocity.
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The rotation is stable with respect to the main axes of inertia, which coincide with the axes of symmetry of the bodies. The rotation is stable with respect to the main axes of inertia, which coincide with the axes of symmetry of the bodies. If at the initial moment the angular velocity deviates slightly in the direction from the axis, which corresponds to the intermediate value of the moment of inertia, then in the future the angle of deviation rapidly increases, and instead of a simple uniform rotation around a constant direction, the body begins to perform a seemingly random somersault.
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Spin plays an important role in team sports: tennis, billiards, baseball. The amazing “dry leaf” kick in football is characterized by a special flight path of a spinning ball due to the occurrence lifting force in the oncoming air flow (Magnus effect). Spin plays an important role in team sports: tennis, billiards, baseball. An amazing “dry leaf” kick in football is characterized by a special flight path of a rotating ball due to the occurrence of lift in the oncoming air flow (Magnus effect).
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The Hubble Space Telescope floats freely in space. How can you change its orientation so as to aim at objects important to astronomers? The Hubble Space Telescope floats freely in space. How can you change its orientation so as to aim at objects important to astronomers?
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Why does a cat always land on its feet when it falls? Why does a cat always land on its feet when it falls? Why is it difficult to maintain balance on a stationary two-wheeled bicycle, and not at all difficult when the bicycle is moving? How will the cockpit of a helicopter in flight behave if, for some reason, the tail rotor stops working?
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In planar motion, the kinetic energy of a rigid body is equal to the sum of the kinetic energy of rotation around an axis passing through the center of mass and the kinetic energy of translational motion of the center of mass: In planar motion, the kinetic energy of a rigid body is equal to the sum of the kinetic energy of rotation around an axis passing through the center of mass and translational energy of the center of mass: The same body can also have potential energy ЕP if it interacts with other bodies. Then the total energy is:
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The kinetic energy of any system of material points is equal to the sum of the kinetic energy of the entire mass of the system, mentally concentrated in its center of mass and moving with it, and the kinetic energy of all material points of the same system in their relative motion with respect to the translationally moving coordinate system with the origin in the center wt. The kinetic energy of any system of material points is equal to the sum of the kinetic energy of the entire mass of the system, mentally concentrated in its center of mass and moving with it, and the kinetic energy of all material points of the same system in their relative motion with respect to the translationally moving coordinate system with the origin in the center wt.
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The dependence of the kinetic energy of rotation on the moment of inertia of bodies is used in inertial batteries. The dependence of the kinetic energy of rotation on the moment of inertia of bodies is used in inertial batteries. The work done due to the kinetic energy of rotation is equal to: Examples: potter's wheels, massive wheels of water mills, flywheels in internal combustion engines. Flywheels used in rolling mills have a diameter of more than three meters and a mass of more than forty tons.
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Problems for self-study Problems for self-solving A ball rolls down an inclined plane of height h = 90 cm. What linear velocity will the center of the ball have at the moment when the ball rolls down the inclined plane? Solve the problem in dynamic and energetic ways. A homogeneous ball of mass m and radius R rolls down without slipping on an inclined plane making an angle α with the horizon. Find: a) the values of the friction coefficient at which there will be no slip; b) the kinetic energy of the ball t seconds after the start of motion.
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“It has long been customary that in a capacitor, this charge keeper, there is an electric field, and in a coil with current, a magnetic field. But to hang a capacitor in a magnetic field - such a thing could only come to the mind of a very Curious child. And not in vain - he learned something new ... It turns out, - the Curious child said to himself, - the electromagnetic field has the attributes of mechanics: the density of momentum and angular momentum! (Stasenko A.L. Why should a capacitor be in a magnetic field? Kvant, 1998, No. 5). “It has long been customary that in a capacitor, this charge keeper, there is an electric field, and in a coil with current, a magnetic field. But to hang a capacitor in a magnetic field - such a thing could only come to the mind of a very Curious child. And not in vain - he learned something new ... It turns out, - the Curious child said to himself, - the electromagnetic field has the attributes of mechanics: the density of momentum and angular momentum! (Stasenko A.L. Why should a capacitor be in a magnetic field? Kvant, 1998, No. 5). “And what do they have in common - rivers, typhoons, molecules?...” (Stasenko A.L. Rotation: rivers, typhoons, molecules. Kvant, 1997, No. 5).
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Read books: Orir D. Popular Physics. M.: Mir, 1964, or Cooper L. Physics for everyone. M .: Mir, 1973. Vol. 1. From them you will learn a lot of interesting things about the movement of planets, wheels, spinning tops, the rotation of a gymnast on the crossbar and ... why a cat always falls on its paws. Read books: Orir D. Popular Physics. M.: Mir, 1964, or Cooper L. Physics for everyone. M .: Mir, 1973. Vol. 1. From them you will learn a lot of interesting things about the movement of planets, wheels, spinning tops, the rotation of a gymnast on the crossbar and ... why a cat always falls on its paws. Read in "Quantum": Vorobyov I. Unusual Journey. (№2, 1974) Davydov V. How do the Indians throw the tomahawk? (№ 11, 1989) Jones D., Why the bicycle is stable (№12, 1970) Kikoin A. Rotational motion of bodies (№1, 1971) Krivoshlykov S. Mechanics of a rotating top. (№ 10, 1971) Lange W. Why the book tumbles (N3,2000) Thomson JJ On the dynamics of a golf ball. (№8, 1990) Use the educational resources of the Internet: http://physics.nad.ru/Physics/Cyrillic/mech.htm http://howitworks.iknowit.ru/paper1113.html http://class-fizika. narod.ru/9_posmotri.htm and others.
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Study the laws of rotational motion using a simulator (Java applet) Study the laws of rotational motion using a simulator (Java applet) FREE ROTATION OF A SYMMETRIC TOP FREE ROTATION OF A HOMOGENEOUS CYLINDER (SYMMETRIC TOP) FORCED PRECESSION OF A GYRO educational resources of the Internet. Perform an experimental study "Determination of the position of the center of mass and moments of inertia of the human body relative to the anatomical axes." Be observant!
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Textbook for grade 10 with an in-depth study of physics, edited by A. A. Pinsky, O. F. Kabardin. M .: "Enlightenment", 2005. Textbook for grade 10 with in-depth study of physics, edited by A. A. Pinsky, O. F. Kabardin. M.: "Enlightenment", 2005. Optional course in physics. O. F. Kabardin, V. A. Orlov, A. V. Ponomareva. M .: "Enlightenment", 1977 Remizov A. N. Physics course: Proc. for universities / A. N. Remizov, A. Ya. Potapenko. M.: Bustard, 2004. Trofimova T. I. Course of physics: Proc. allowance for universities. Moscow: Vysshaya Shkola, 1990. http://ru.wikipedia.org/wiki/ http://elementy.ru/trefil/21152 http://www.physics.ru/courses/op25part1/content/chapter1/section /paragraph23/theory.html Physclips. Multimedia introduction to physics. http://www.animations.physics.unsw.edu.au/jw/rotation.htm and others. Illustrative materials from the Internet were used in the design for educational purposes.
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