MAHA CONE- conical a surface that limits an area in a supersonic gas flow in which M. oscillates under the influence of. The simplest M. consists of a small massive load WITH, suspended on a thread (or light rod) length l. If we consider the thread to be inextensible and neglect the size of the load compared to the length of the thread, and the mass of the thread compared to the mass of the load, then the load on the thread can be considered as material point, located at a constant distance l from the suspension point ABOUT. (Fig. 1, A).

Such M. is called. circular math. M. If, as is usually the case, the oscillating body cannot be considered as a material point, then M. is called. physical.

Mathematical pendulum (circular). If M., deviated from the equilibrium position C 0, release without starting. speed or inform the point WITH speed perpendicular OS and lying in the plane of the beginning. deviations, then M. will oscillate in one rotation. plane (plane math. M.). If we neglect friction in the axis and air resistance (which is always assumed in the future), then for M. the law of conservation of mechanical forces will apply. energy, which gives:

where is the speed of the point WITH, - its coordinate, measured vertically upward from the equilibrium position, - - the angle of deviation M. from the vertical, g- acceleration of gravity, h- constant, proportional to the total mechanical energy M. and determined beginning. values

When the reported M. began. the energy is such that (for a load on a rod) or (for a load on a thread), then M. will oscillate with an angle. amplitude determined by the equality These oscillations are not harmonic; their period T depends on amplitude and the trace is determined, the formula obtained from equation (1):

When the above conditions for k are not fulfilled, then M. does not oscillate. movements. For example, when the load on the rod will describe a circle. When the reported M. began. energy is very low M. performs small oscillations, close to harmonic; The period of small oscillations can be approximately considered equal to:

i.e., independent of amplitude (oscillations are isochronous). F-la (3) compared to (2) gives an error of up to 0.05% at up to 1% at. These results

tats are fair for inertial reference system. In relation to the Earth due to its daily rotation the swing plane M. slowly changes its direction (see. Foucault pendulum).

If the rejected M. is informed of the beginning. speed not lying in the initial plane. deviations, then point WITH will describe on a sphere of radius l curves enclosed between 2 parallels, where the values ​​depend on the beginning. conditions (spherical M., Fig. 2, i). In the special case, at point WITH will describe a horizontal circle (conical M., Fig. 2, b). From non-circular M. special interest is cycloidal pendulum, the oscillations of which are isochronous at any amplitude.

Physical pendulum. Phys. M. usually called a solid body that, under the influence of gravity, oscillates around the horizontal axis of the suspension (Fig. 1, b). The motion of such a M. is quite similar to the motion of a circular mat. M. Period of finite or small oscillations of physical. M. is determined accordingly by formulas (2) or (3), in which l should be replaced by the value where T- mass M., a - distance from the center of gravity WITH to the suspension axis, I- M. relative to the suspension axis, - relative to the axis, parallel axis suspension and passing through WITH. The period depends on the position of the suspension axis relative to the center of gravity and will be smallest at Value l 0, there is always more of heaven A, called given length of physical M. If laid along the line OS line segment OK = l 0, then the resulting point K groove. center of physical swings M. (math. M. with mass concentrated at a point TO, will fluctuate with the same period as the given physical. M.). Gimbal axis point ABOUT and swing center K have the property of reciprocity: if M. is suspended so that the axis of the suspension passes through K, then point ABOUT will become the center of oscillation and the period of oscillation M. will not change. This property is the basis of the reverse M device, which is used to determine the acceleration of gravity.

The properties of M. are widely used in various industries. devices: watches, devices for determining the acceleration of gravity (pendulum device), acceleration of moving bodies, oscillations earth's crust(seismograph), in gyroscopic. devices, devices for experiments. determining the moments of inertia of bodies, etc.

Mach cone

(named after E. Mach), characteristic cone, is a region of the supersonic flow field in which infinitesimal pressure disturbances propagate from a point source of disturbances. If a point source of disturbances P moves in a homogeneous compressible medium with a supersonic speed V, then during its movement it generates infinitesimal pressure disturbances that propagate in the medium with the speed of sound a. Since V > a, the disturbances caused by the source P in positions 3, 2, 1 and 0 cannot catch up and overtake the source P in position 4. Thus, all disturbances will propagate in the flow inside a cone with a vertex at point P and an angle half-solution (), called the Mach angle and determined by the formula
sin(() = a/V = 1/M,
where M is the Mach number. The PA line is called the Mach line or wave; it is the envelope surface of the leading edges sound waves, and on it the disturbances are most densely located, since all sound waves are in the same oscillation phase - in the compression phase. The surface of the magnetic field serves as a natural boundary dividing the entire space into two regions - undisturbed and disturbed. This concentration of disturbances inside the airframe determines many features of the aerodynamics of high speeds.
The source of small disturbances can practically be any small obstacle, for example a mark or a bump on a streamlined surface. In a supersonic flow, a Mach wave departs from each small obstacle, limiting the area of ​​propagation of disturbances caused by this obstacle. This property is used in optical methods studying flows around a body by marking its surface.

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"Mach cone" in books

Cone

From the book Testers author Vishenkov S

Cone In the test pilots' newspaper it was printed in large letters: “The everyday life of flight test work is permeated with courage, endurance, composure, perseverance, skill.” These words were very often justified in practice. And the event that took place in the cool summer

Cone of force

From the book Young Sorceress, or Magic for Teenagers author Ravenwolf Silver

Cone of Power You will often hear witches talk about the “cone of power.” This cone of energy occurs when one witch (or group of witches) performs a ritual or performs magic. Imagine a group of people standing in a circle. Those who dance well take places

Cone

From the book Encyclopedic Dictionary (K) author Brockhaus F.A.

Cone Cone (straight circular) – geometric body, formed by rotation right triangle near one of the legs. The hypotenuse is called the generator; fixed leg – height; a circle described by a rotating leg - the base. Side surface TO.

Cone

From the book Big Soviet Encyclopedia(KO) of the author TSB

Mach cone

From the book Great Soviet Encyclopedia (MA) by the author TSB

Cone

From the book AutoCAD 2009 for students. Self-instruction manual author Sokolova Tatyana Yurievna

Cone The CONE command forms a solid cone (Fig. 16.6), the base of which (circle or ellipse) lies in the XY plane of the current coordinate system, and the vertex is located along the Z axis. The command is called from the Draw ? Modeling? Cone, or by clicking the Cone icon on the panel

Cone

From the book AutoCAD 2010 author Orlov Andrey Alexandrovich

Cone Using the AI_CONE command, you can build a full or truncated cone (Fig. 10.11). Rice. 10.11. ConesThe first step is to specify the center point of the base of the cone in response to the prompt:Specify center point for base of cone:Then the following prompt appears:Specify radius for base of cone or :In response

Cone

From the book AutoCAD 2010 author Orlov Andrey Alexandrovich

Cone The CONE command allows you to construct a straight cone with a circle or ellipse at the base. This command also makes it possible to create not only a full, but also a truncated cone (Fig. 11.3). Rice. 11.3. Cone parametersTo start building a cone, run the command

Cone

From the book AutoCAD 2008 for students: a popular tutorial author Sokolova Tatyana Yurievna

Cone The CONE command forms a solid cone, the base of which (circle or ellipse) lies in the XY plane of the current coordinate system, and the vertex is located along the Z axis. The command is called from the drop-down menu Draw ? Modeling? Cone or by clicking the Cone icon in the panel

Cone

author Sokolova Tatyana Yurievna

Cone Command AI_CONE, forms a surface circular cone. Team Requests

Cone

From the book AutoCAD 2009. Training course author Sokolova Tatyana Yurievna

Cone The CONE command forms a solid cone, the base of which (circle or ellipse) lies in the XY plane of the current coordinate system, and the vertex is located along the Z axis. Is the command called from the Draw drop-down menu? Modeling? Cone, or by clicking the Cone icon on the panel

Cone

From the book AutoCAD 2009 author Orlov Andrey Alexandrovich

Cone Using the AI_CONE command, you can build a full or truncated cone (Fig. 10.11). Rice. 10.11. ConesThe first step is to specify the center of the base of the cone in response to the prompt: Specify center point for base of cone: After this, the following prompt appears: Specify radius for base of cone or : In response

Cone

From the book AutoCAD 2009 author Orlov Andrey Alexandrovich

Cone The CONE command allows you to construct a straight cone with a circle or ellipse at the base. This command has been significantly changed compared to previous versions of the program, and, for example, now it allows you to create not only a full, but also a truncated cone (Fig.

Cone

From the book AutoCAD 2009. Let's get started! author Sokolova Tatyana Yurievna

Cone The CONE command forms a solid cone, the base of which (circle or ellipse) lies in the XY plane of the current coordinate system, and the vertex is located along the Z axis. The command is called from the Draw ? Modeling? Cone, or by clicking the Cone icon on the panel

Maha Bandha, Maha Vedha and Maha Mudra are the keys to the practice of transforming sexual energy into vitality

From the book Yoga Therapy. A New Look for traditional yoga therapy author Sivananda Swami

Maha-bandha, Maha-vedha and Maha-mudra are the keys to the practice of transforming sexual energy into vitality Maha-bandha, Maha-vedha and Maha-mudra are three very special (exercises that are always described together in all yoga texts and are, in fact,

If you have ever stood next to a flying supersonic aircraft, you probably remember the deafening sound of the shock wave that accompanies the movement of a body at a speed of more than Mach 1, that is, greater than the speed of sound in a given environment. The area of ​​propagation of the shock wave from a supersonic aircraft is limited by the Mach cone. A group of scientists from the University of Illinois at Urbana-Champaign (USA) and Tsinghua Research University (China) managed to capture on camera for the first time “ shock wave"from photons. Like sound, photons of light have a wave nature, and therefore form the same Mach cone if the body moves faster than the speed of light in the environment.

Mach sound cone

A Mach cone occurs when a body moves faster than the waves it generates. Most often they talk about a sonic shock wave from an aircraft that flies at a speed of more than Mach 1, that is, greater than the speed of sound in a given environment.

In general, when moving at transonic speeds, whole line interesting effects, including the Prandtl-Gloert effect: a beautiful cloud behind the plane.


Prandtl-Gloert effect: a phenomenon consisting of condensation atmospheric moisture behind an object moving at transonic speeds

A cloud occurs when an airplane flying at high speed creates an area of ​​low pressure behind it. After the flight, this area is filled with surrounding air, during which the air temperature drops sharply below the dew point (a temperature jump as a result of an adiabatic process). If the air humidity is high, then water vapor condenses into tiny droplets that form a cloud.

The propagation of a sound shock wave is also an adiabatic process, like the Prandtl-Gloert effect. Here, in air environment there is a jump in pressure, density, temperature and air speed. Sound itself is fluctuations in the density, speed and pressure of a medium. Adiabatic process at supersonic speed it is accompanied by a shock wave, which at a distance from the energy source degenerates into a sound wave, and the speed of its propagation approaches the speed of sound.

The Prandtl-Gloert cloud shown above is not directly associated with the shock wave. It occurs simply due to cooling of the air and the formation of condensation. That is, this process cannot be called “visualization” of the Mach cone. But an experiment by scientists from the University of Illinois at Urbana-Champaign and Tsinghua University is a direct observation of this effect. Not for sound, but for light.

Mach light cone

The light shock wave also has the shape of a cone, just like the sound shock wave. To record it on video, the researchers used laser pulses as a moving body. They used a clever technique in which pulses of light travel at “superluminal” speeds, that is, faster than the speed of light in the environment.

The first task in this experiment was to slow down the light. Everyone knows that the speed of light in a vacuum is about 300,000 km/s, but in other media light moves more slowly, until it stops completely. To slow down light in this experiment, scientists filled carbon dioxide a tunnel between two plates made from a mixture of silicone rubber and aluminum oxide powder.

Pulses were launched into this tunnel green laser lasting 7 picoseconds. The trick is that photons move faster inside the tunnel than through the plates along the tunnel. Therefore, when moving through the tunnel, laser pulses left behind a conical trail of slower light waves, which, as a result of scattering, overlapped each other in the plates - this is the Mach cone.

In previous years, experiments had already been carried out that recorded the presence of photon Mach cones, but now for the first time, scientists were able to film in real time on a video camera how a single laser pulse moves in space.

To do this, it was necessary to construct a special electron-optical camera (slit camera), which can take up to 100 billion frames per second in one exposure. The camera operated in three modes: the first one filmed the phenomenon itself, and the other two recorded information about time. These data were then combined to produce a scientifically reliable video recording of the propagation of the photon Mach cone.
An electron-optical camera of this design can be used in medicine and other fields of science to record unpredictable light phenomena. Unlike other cameras, there is no pre-setup or thousands of individual frames required. This camera operates at one shutter speed.
The authors suggest that this camera can be used to videotape the impulses that neurons exchange with each other during the process. mental activity. It becomes possible to accurately record electronic traffic in the human brain. “We hope that we can use our system to study neural networks to understand how the brain works,” said optical engineer Jinyang Liang of Washington University in St. Louis, lead author of the paper.

Research Article

Mach cone- a conical surface that limits the region in a supersonic gas flow in which disturbances (sound waves) generated by point source disturbances - a body streamlined by a flow or, equivalently, moving in a medium at supersonic speed; the Mach cone delimits the disturbed and undisturbed regions of the medium. Named after Ernst Mach, who introduced this concept into physics.

The surface of the Mach cone is envelope systems of sound waves generated by a body when moving in a medium: in accordance with Huygens' principle the surface of the cone is formed by the interference of sound waves during their superposition and the vibrations on the surface are in one phase - the compression phase, forming shock wave.

The angle between the generators of the cone and its axis is called Mach angle, it is associated with Mach number with the following ratio:

\sin \alpha = \frac(c t)(v t) = \frac(c)(v) = \frac(1)(M)

Where: $\backslash alpha$: Mach angle (φ in the picture) $c$: sound speed $v$: flow rate $M$: Mach number

In electrodynamics, the Mach cone corresponds to the “Cherenkov cone” - a conical envelope Cherenkov radiation, arising when moving in a medium elementary particle at a speed exceeding the speed of light propagation in the medium.

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Excerpt characterizing the Mach Cone

- ABOUT! what are you saying? said another. -Where will he go? It's closer here.
Rostov thought about it and drove exactly in the direction where he was told that he would be killed.
“Now it doesn’t matter: if the sovereign is wounded, should I really take care of myself?” he thought. He entered the area where most of the people fleeing from Pracen died. The French had not yet occupied this place, and the Russians, those who were alive or wounded, had long abandoned it. On the field, like heaps of good arable land, lay ten people, fifteen killed and wounded on every tithe of space. The wounded crawled down in twos and threes together, and one could hear their unpleasant, sometimes feigned, as it seemed to Rostov, screams and moans. Rostov started to trot his horse so as not to see all these suffering people, and he became scared. He feared not for his life, but for the courage that he needed and which, he knew, would not withstand the sight of these unfortunates.
The French, who stopped shooting at this field strewn with the dead and wounded, because there was no one alive on it, saw the adjutant riding along it, aimed a gun at him and threw several cannonballs. The feeling of these whistling, terrible sounds and the surrounding dead people merged for Rostov into one impression of horror and self-pity. He remembered last letter mother. “What would she feel,” he thought, “if she saw me now here, on this field and with guns pointed at me.”
In the village of Gostieradeke there were, although confused, more order Russian troops marching away from the battlefield. The French cannonballs could no longer reach here, and the sounds of firing seemed distant. Here everyone already clearly saw and said that the battle was lost. To whomever Rostov turned, no one could tell him where the sovereign was, or where Kutuzov was. Some said that the rumor about the sovereign’s wound was true, others said that it was not, and explained this false rumor that had spread by the fact that, indeed, the pale and frightened Chief Marshal Count Tolstoy galloped back from the battlefield in the sovereign’s carriage, who rode out with others in the emperor’s retinue on the battlefield. One officer told Rostov that beyond the village, to the left, he saw someone from the higher authorities, and Rostov went there, no longer hoping to find anyone, but only to clear his conscience before himself. Having traveled about three miles and having passed the last Russian troops, near a vegetable garden dug in by a ditch, Rostov saw two horsemen standing opposite the ditch. One, with a white plume on his hat, seemed familiar to Rostov for some reason; another, unfamiliar rider, on a beautiful red horse (this horse seemed familiar to Rostov) rode up to the ditch, pushed the horse with his spurs and, releasing the reins, easily jumped over the ditch in the garden. Only the earth crumbled from the embankment from the horse’s hind hooves. Turning his horse sharply, he again jumped back over the ditch and respectfully addressed the rider with the white plume, apparently inviting him to do the same. The horseman, whose figure seemed familiar to Rostov and for some reason involuntarily attracted his attention, made a negative gesture with his head and hand, and by this gesture Rostov instantly recognized his lamented, adored sovereign.