Maxwell's magnetic wheel
Content
The English physicist James Clark Maxwell, who lived from 1831-79, is best known for formulating the system of equations underlying electrodynamics—and using it to predict the existence of electromagnetic waves. However, this is not all of his significant achievements. Maxwell was also involved in thermodynamics, incl. gave the concept of the famous "demon" that directs the movement of gas molecules, and derived a formula describing the distribution of their velocities. He also studied color composition and invented a very simple and interesting device to demonstrate one of the most basic laws of nature - the principle of conservation of energy. Let's try to get to know this device better.
The mentioned apparatus is called Maxwell's wheel or pendulum. We will deal with two versions of it. First will be invented by Maxwell - let's call it classic, in which there are no magnets. Later we will discuss the modified version, which is even more amazing. Not only will we be able to use both demo options, i.e. quality experiments, but also to determine their effectiveness. This size is an important parameter for every engine and working machine.
Let's start with the classic version of Maxwell's wheel.
Lynx. one. The classic version of Maxwell's wheel: 1 - horizontal bar, 2 - strong thread, 3 - axle, 4 - wheel with a high moment of inertia.
The classic version of the Maxwell wheel is shown in Fig. fig. 1. To make it, we attach a strong rod horizontally - it can be a stick-brush tied to the back of a chair. Then you need to prepare a suitable wheel and put it motionless on a thin axle. Ideally, the diameter of the circle should be approximately 10-15 cm, and the weight should be approximately 0,5 kg. It is important that almost the entire mass of the wheel falls on the circumference. In other words, the wheel should have a light center and a heavy rim. For this purpose, you can use a small spoked wheel from a cart or a large tin lid from a can and load them around the circumference with the appropriate number of turns of wire. The wheel is placed motionless on a thin axle at half of its length. The axis is a piece of aluminum pipe or rod with a diameter of 8-10 mm. The easiest way is to drill a hole in the wheel with a diameter of 0,1-0,2 mm less than the diameter of the axle, or use an existing hole to put the wheel on the axle. For a better connection with the wheel, the axle can be smeared with glue at the point of contact of these elements before pressing.
On both sides of the circle, we tie segments of a thin and strong thread 50-80 cm long to the axis. However, a more reliable fixation is achieved by drilling the axis at both ends with a thin drill (1-2 mm) along its diameter, inserting a thread through these holes and tying it. We tie the remaining ends of the thread to the rod and thus hang the circle. It is important that the axis of the circle is strictly horizontal, and the threads are vertical and evenly spaced from its plane. For completeness of information, it should be added that you can also buy a finished Maxwell wheel in companies that sell teaching aids or educational toys. In the past, it was used in almost every school physics lab.
First experiments
Let's start with the situation when the wheel hangs on the horizontal axis in the lowest position, i.e. both threads are completely unwound. We grasp the axle of the wheel with our fingers at both ends and slowly rotate it. Thus, we wind the threads on the axis. You should pay attention to the fact that the next turns of the thread are evenly spaced - one next to the other. The wheel axle must always be horizontal. When the wheel approaches the rod, stop winding and let the axle move freely. Under the influence of weight, the wheel begins to move downward and the threads unwind from the axle. The wheel spins very slowly at first, then faster and faster. When the threads are fully unfolded, the wheel reaches its lowest point, and then something amazing happens. The rotation of the wheel continues in the same direction, and the wheel begins to move upward, and threads are wound around its axis. The speed of the wheel gradually decreases and eventually becomes equal to zero. The wheel then appears to be at the same height as before it was released. The following up and down movements are repeated many times. However, after a few or a dozen such movements, we notice that the heights to which the wheel rises become smaller. Eventually the wheel will stop in its lowest position. Before this, it is often possible to observe oscillations of the axis of the wheel in a direction perpendicular to the thread, as in the case of a physical pendulum. Therefore, Maxwell's wheel is sometimes called a pendulum.
Lynx. one. The main parameters of the Maxwell wheel: - weight, - wheel radius, - axle radius, - weight of the wheel with the axle, - linear speed, 0 - initial height.
Let us now explain why the Maxwell wheel behaves in this way. Winding the threads on the axle, raise the wheel in height 0 and do work through it (fig. 2). As a result, the wheel in its highest position has the potential energy of gravity pexpressed by the formula [1]:
where is the free fall acceleration.
As the thread unwinds, the height decreases, and with it the potential energy of gravity. However, the wheel picks up speed and thus acquires kinetic energy. kwhich is calculated by the formula [2]:
where is the moment of inertia of the wheel, and is its angular velocity (= /). In the lowest position of the wheel (0 = 0) the potential energy is also equal to zero. This energy, however, did not die, but turned into kinetic energy, which can be written according to the formula [3]:
As the wheel moves up, its speed decreases, but the height increases, and then the kinetic energy becomes potential energy. These changes could take any amount of time if it were not for the resistance to movement - air resistance, resistance associated with the winding of the thread, which require some work and cause the wheel to slow down to a complete stop. The energy does not press, because the work done in overcoming the resistance to motion causes an increase in the internal energy of the system and an associated increase in temperature, which could be detected with a very sensitive thermometer. Mechanical work can be converted into internal energy without limitation. Unfortunately, the reverse process is constrained by the second law of thermodynamics, and so the potential and kinetic energy of the wheel eventually decrease. It can be seen that Maxwell's wheel is a very good example to show the transformation of energy and explain the principle of its behavior.
Efficiency, how to calculate it?
The efficiency of any machine, device, system or process is defined as the ratio of energy received in useful form. u to delivered energy d. This value is usually expressed as a percentage, so the efficiency is expressed by the formula [4]:
.
The efficiency of real objects or processes is always below 100%, although it can and should be very close to this value. Let us illustrate this definition with a simple example.
The useful energy of an electric motor is the kinetic energy of rotational motion. In order for such an engine to work, it must be powered by electricity, for example, from a battery. As you know, part of the input energy causes heating of the windings, or is needed to overcome the friction forces in the bearings. Therefore, the useful kinetic energy is less than the input electricity. Instead of energy, the values of [4] can also be substituted into the formula.
As we established earlier, Maxwell's wheel has the potential energy of gravity before it starts to move. p. After completing one cycle of up and down motions, the wheel also has gravitational potential energy, but at a lower height. 1so there is less energy. Let's denote this energy as P1. According to the formula [4], the efficiency of our wheel as an energy converter can be expressed by the formula [5]:
Formula [1] shows that potential energies are directly proportional to height. When substituting formula [1] into formula [5] and taking into account the corresponding height marks and 1, then we get [6]:
Formula [6] makes it easy to determine the efficiency of the Maxwell circle - it is enough to measure the corresponding heights and calculate their quotient. After one cycle of movements, the heights can still be very close to each other. This can happen with a carefully designed wheel with a large moment of inertia raised to a considerable height. So you will have to take measurements with great accuracy, which will be difficult at home with a ruler. True, you can repeat the measurements and calculate the average, but you will get the result faster after deriving a formula that takes into account growth after more movements. When we repeat the previous procedure for driving cycles, after which the wheel will reach its maximum height n, then the efficiency formula will be [7]:
height n after a few or a dozen or so cycles of movement, it is so different from 0that it will be easy to see and measure. The efficiency of the Maxwell wheel, depending on the details of its manufacture - size, weight, type and thickness of the thread, etc. - is usually 50-96%. Smaller values are obtained for wheels with small masses and radii suspended on stiffer threads. Obviously, after a sufficiently large number of cycles, the wheel stops in the lowest position, i.e. n = 0. The attentive reader, however, will say that then the efficiency calculated by formula [7] is equal to 0. The problem is that in the derivation of formula [7], we tacitly adopted an additional simplifying assumption. According to him, in each cycle of movement, the wheel loses the same share of its current energy and its efficiency is constant. In the language of mathematics, we assumed that successive heights form a geometric progression with a quotient. In fact, this should not be until the wheel finally stops at a low height. This situation is an example of a general pattern, according to which all formulas, laws and physical theories have a limited scope of applicability, depending on the assumptions and simplifications adopted in their formulation.
Magnetic version
Lynx. one. Maxwell's magnetic wheel: 1 - a wheel with a high moment of inertia, 2 - an axis with magnets, 3 - a steel guide, 4 - a connector, 5 - a rod.
Now we will deal with the magnetic version of the Maxwell wheel - construction details are presented Rice. 3 and 4. To assemble it, you will need two cylindrical neodymium magnets with a diameter of 6-10 mm and a length of 15-20 mm. We will make the wheel axle from an aluminum tube with an inner diameter equal to the diameter of the magnets. The wall of the tube should be thin enough
1 mm. We insert the magnets into the tube, placing them at a distance of 1-2 mm from its ends, and glue them with epoxy glue, such as Poxipol. The orientation of the poles of the magnets does not matter. We close the ends of the tube with small aluminum discs, which will make the magnets invisible, and the axis will look like a solid rod. The conditions to be met by the wheel and how to install it are the same as before.
For this version of the wheel, it is also necessary to make steel guides from two sections installed in parallel. An example of the length of guides that are convenient in practical use is 50-70 cm. The so-called closed profiles (hollow inside) of square section, the side of which has a length of 10-15 mm. The distance between the guides must be equal to the distance of the magnets placed on the axis. The ends of the guides on one side should be filed in a semicircle. For better retention of the axis, pieces of a steel rod can be pressed into the guides in front of the file. The remaining ends of both rails must be attached to the rod connector in any way, for example, with bolts and nuts. Thanks to this, we got a comfortable handle that can be held in your hand or attached to a tripod. The appearance of one of the manufactured copies of Maxwell's magnetic wheel shows PHOT. one.
To activate Maxwell's magnetic wheel, place the ends of its axle against the top surfaces of the rails near the connector. Holding the guides by the handle, tilt them diagonally towards the rounded ends. Then the wheel begins to roll along the guides, as if on an inclined plane. When the round ends of the guides are reached, the wheel does not fall, but rolls over them and
Lynx. one. Details of the design of Maxwell's magnetic wheel are shown in axial section:
1 - wheel with a high moment of inertia, 2 - aluminum tube axle, 3 - cylindrical neodymium magnet, 4 - aluminum disk.
it makes an amazing evolution - it rolls up the lower surfaces of the guides. The described cycle of movements is repeated many times, like the classical version of Maxwell's wheel. We can even set the rails vertically and the wheel will behave exactly the same. Keeping the wheel on the guide surfaces is possible due to the attraction of the axle with neodymium magnets hidden in it.
If, at a large angle of inclination of the guides, the wheel slides along them, then the ends of its axis should be wrapped with one layer of fine-grained sandpaper and glued with Butapren glue. In this way, we will increase the friction necessary to ensure a rolling without slipping. When the magnetic version of the Maxwell wheel moves, similar changes in mechanical energy occur, as in the case of the classical version. However, the energy loss may be somewhat greater due to friction and magnetization reversal of the guides. For this version of the wheel, we can also determine the efficiency in the same way as described earlier for the classic version. It will be interesting to compare the obtained values. It is easy to guess that the guides do not have to be straight (they can be, for example, wavy) and then the movement of the wheel will be even more interesting.
and energy storage
The experiments carried out with the Maxwell wheel allow us to draw several conclusions. The most important of these is that energy transformations are very common in nature. There are always so-called energy losses, which are actually transformations into forms of energy that are not useful for us in a given situation. For this reason, the efficiency of real machines, devices and processes is always less than 100%. That is why it is impossible to build a device that, once set in motion, will move forever without an external supply of energy necessary to cover the losses. Unfortunately, in the XNUMXth century, not everyone is aware of this. That is why, from time to time, the Patent Office of the Republic of Poland receives a draft invention of the type “Universal device for driving machines”, using the “inexhaustible” energy of magnets (probably happens in other countries as well). Of course, such reports are rejected. The rationale is short: the device will not work and is not suitable for industrial use (therefore does not meet the necessary conditions for obtaining a patent), because it does not comply with the basic law of nature - the principle of conservation of energy.
Photo 1. Appearance of one of Maxwell's magnetic wheels.
Readers may notice some analogy between Maxwell's wheel and the popular toy called the yo-yo. In the case of the yo-yo, the loss of energy is replenished by the work of the user of the toy, who rhythmically raises and lowers the upper end of the string. It is also important to conclude that a body with a large moment of inertia is difficult to rotate and difficult to stop. Therefore, Maxwell's wheel slowly picks up speed when moving down and also slowly decreases it as it goes up. The up and down cycles are also repeated for a long time before the wheel finally stops. All this is because a large kinetic energy is stored in such a wheel. Therefore, projects are being considered for the use of wheels with a large moment of inertia and previously brought into very fast rotation, as a kind of "accumulator" of energy, intended, for example, for additional movement of vehicles. In the past, powerful flywheels were used in steam engines to provide more even rotation, and today they are also an integral part of automobile internal combustion engines.
