METHODOLOGICAL NOTES

SEVERAL DEMONSTRATION EXPERIMENTS FOR A GENERAL PHYSICS COURSE

M. A. Grabovskii and V. S. Egorov

1. Apparatus for demonstrating the motion of the center of mass of a system—“The Falling Board.” It is known that the center of mass of any mechanical system moves exactly as a material point in which the mass of the entire system is concentrated and to which all external forces acting on the system are applied.

Fig. 1.

To demonstrate this fundamental proposition of the dynamics of a system, the following experiment is proposed. A board whose center of gravity is known is placed in an inclined position and held there with the aid of an electromagnet (Fig. 1). At the left end of the board an axle \(P\) is fixed, on which two ball bearings are pressed. In this case, when the board falls, its lower end will be

roll along the horizontal support of the apparatus, and the friction forces developing here, as experience shows, are small and may be neglected. Consequently, only two vertical forces will act on the falling board: the force of gravity and the reaction of the support, directed upward at the point where the board touches the rollers; these forces determine the motion of the system’s center of mass in the vertical direction.

To determine the trajectory of the motion of the system’s center of mass, two paint-covered brushes are attached to the side of the board; they lightly touch a white sheet of paper fastened to the vertical shield of the apparatus. The first brush is attached at the center of gravity of the board, while the second brush is somewhat higher, closer to the end of the board held by the electromagnet.

If the current in the winding of the electromagnet is now switched off, the board will fall, and its lower end will roll with very little friction along the horizontal support of the apparatus. The first brush, fixed at the center of mass of the board, will draw a straight line on the paper; the second brush, corresponding in this case to an arbitrary point of the system, will trace a concave curve, turned with its curvature toward the center of mass of the board. The curved line (part of an ellipse) drawn by the second brush convinces the observer that all points of the system, except the center of mass, move not rectilinearly, but curvilinearly, rotating about the falling center of mass of the board.

The experiment must be repeated in a second version. The board is again placed obliquely. This time the board is loaded with an additional weight (approximately \(1\) kg), which is fastened near the upper end of the board. Now the center of mass of the system rises upward, and its new position approximately coincides with the place where the second brush is attached (see Fig. 1). When the board falls in this case, the second brush will draw a straight line, while the first brush, no longer connected with the center of gravity of the system, will draw a curved line, turned with its concavity toward the new position of the center of mass of the system. Additional weights may be attached at different places on the board, thereby repeating the experiment in other variants.

Practical instructions: 1. All parts of the installation for demonstrating the motion of the center of mass are mounted on a stable vertical frame \(A\). A smooth wooden board \(B\) is attached to the base of the frame (Fig. 1). The falling board has approximately the following dimensions: length \(140\) cm, width \(90\) mm, and thickness \(30\) mm. In accordance with the dimensions of the falling board, the dimensions of the supporting frame are chosen. The frame of the apparatus is covered with a wooden shield, to which sheets of paper are attached during the experiment.

Fig. 2.

1. An electromagnet is attached to the upper base of the apparatus (see Figs. 1, 2). To the raised upper end of the board is attached a small soft-iron core \(C\), provided with two brass projections of spherical form (Fig. 2). These projections enter recesses made in the ends of the iron core of the electromagnet. In this case the initial position of the “falling board” is strictly fixed, and it is easy to set the board so that the brushes lightly touch the paper, while the motion of the whole board, when the current in the electromagnet winding is switched off, would always occur in a definite direction. It is recommended to switch off the current in the electromagnet winding by means of a push-button switch.

2. Before the demonstration of each variant of the experiment, it is necessary to show where the center of mass of the system is located. Since the position of the first brush corresponds, as we noted above, to the center of mass of the unloaded board, and the position of the second brush to that of the loaded board, hooks are provided in the construction of the apparatus. Having tied a thread to the hook, the experimenter, with the board raised in the horizontal position, checks before the spectators the position of the center of mass of the board in both cases.

3. In order to eliminate bouncing of the falling board when it strikes the support, which may cause distortion of the lines drawn on the paper and the board to fall onto the floor, the apparatus is provided with a trap \(T\), fastened to the base of the apparatus. The “trap” consists of two fairly large springs made of work-hardened brass, between which there is a cushion stuffed with wadding. In this case the falling board drops into the trap, and bouncing of the board is completely eliminated. A safety bracket \(D\), fastened to the base of the apparatus, serves the same purpose.

2. Resonance of a Motor Suspended on a Spring

In engineering there are known cases when a rotating engine or motor, at resonance, shakes loose the foundation on which it is mounted. Resonance is dangerous especially at the frequency of the motor’s separate parts of a flying airplane or of the entire machine as a whole. There are also known cases of bridges being destroyed under the action of periodic jolts at the joints of trains moving over them, or under the action of soldiers rhythmically marching across a bridge. This whole range of questions is illustrated by the following demonstration.

A small DC motor is suspended from a massive stand on a cylindrical spring (Fig. 3). A small rod is fastened asymmetrically to the motor shaft to create jolts, whose period corresponds to the rotational frequency of the motor. At a small number of revolutions, approximately 85 rev/min, the first resonance appears, expressed in the periodic lowering and rising of the motor. This resonance is determined by longitudinal deformation of the spring. The frequency at which it occurs depends on the elastic coefficient of the spring and on the mass of the motor. When the number of revolutions of the suspended motor is increased, the first resonance disappears, and at frequencies of approximately 170 rev/min a new resonance appears, determined by transverse oscillations of the spring. The motor begins to oscillate intensely about a horizontal axis passing approximately through its center of mass (Fig. 4).

Fig. 3.          Fig. 4.          Fig. 5.

With a further increase in the number of revolutions, the second resonance disappears, and at several hundred revolutions of the motor per minute there appears, successively, first the third, and then the fourth and even higher-order resonances, associated with the emergence of standing transverse waves on the cylindrical spring itself.

The picture of this last resonance is interesting. The rotating motor remains motionless, while the spring itself enters into intense oscillation and along its length one, two, or more half-waves are established (Fig. 5).

The resonances obtained with such an apparatus are very sharp, and with a sudden change in the number of revolutions they can be missed. Therefore several rheostats (with large and small resistance) must be included in the motor circuit, so that the number of revolutions can be regulated very smoothly.

When demonstrating the experiment in a large auditorium, it is recommended to project the suspended motor in shadow projection onto a screen.

Approximate data for the apparatus:

a) DC motor with a power of ≈ 15 W, voltage 120 V, motor weight 900 g.

b) The cylindrical spring is wound from steel wire; wire diameter ≈ 2.4 mm, diameter of an individual coil of the spring ≈ 22 mm, total length of the unloaded spring ≈ 185 mm; when the motor is suspended, the spring lengthens by 160 mm; number of coils of the spring ≈ 70–75.

c) The current supply to the motor from the DC source is made by a flexible cord.

3. Phenomena of Acoustic Resonance on Helmholtz Resonators

Four Helmholtz resonators of different sizes are arranged in a single line on the demonstration table so that the small, slightly elongated openings of the resonators are located at the same height (Fig. 6). In front of these openings are placed four identical paper spinners (Fig. 7).

On the side of the large openings of the resonators there is a reproducer connected to an audio-frequency generator. By exciting the reproducer with alternating currents of different frequency and thereby creating strong sound fields, one can obtain the phenomenon of resonance, which is convincingly manifested in the fact that one of the four pinwheels rotates—namely the one that stands at the excited resonator. By changing the sound frequency, one can make all four pinwheels rotate in turn.

The experiment is demonstrated in the resonance regime; as soon as the frequency at which the paper wheel rotates is decreased or increased by several percent, in a few seconds it will come to a stop. Therefore it is recommended to change the sound frequency very slowly, stopping for some time at those frequencies at which one or another resonator is excited, as well as at intermediate frequencies when none of the pinwheels rotates.

Fig. 6.

Fig. 7.

For carrying out this experiment, resonators of the following dimensions were used: the diameter of the first resonator was 60 mm, of the second 100 mm, of the third 140 mm, and of the fourth 250 mm. The frequencies at which they were excited were, respectively: 800, 400, 300, and 120 cps. The audio-frequency generator is of type ЗГ-2А. The paper pinwheel has the form of a small wheel (diameter 40–50 mm and height 7–10 mm) resting on a needle (Fig. 7). Between its upper and lower surfaces are glued 8–10 paper blades, arranged at an angle of 30–35° to the direction of the wheel radius. With this arrangement of the blades relative to the air stream issuing from the resonator, one can count on the maximum ponderomotive action of the sound.

To eliminate friction when the pinwheel rotates, a small glass tube sealed at one end is glued into its center. Into the opening of this tube enters the metal needle of the support (Fig. 7). The total weight of the pinwheel is 2.5–3.5 g.

The following variant of the experiment is possible. Instead of Helmholtz resonators, one may take conical resonators made of sheet metal or of cardboard. By exciting the conical resonators in the same manner, one can likewise observe the phenomenon of resonance from the rotation of the paper cylinder. In this case the resonance is less sharp. However, in this form the experiment makes it possible to observe a series of overtones if the frequency of the tone exciting the conical resonator is smoothly increased.

The length of the largest conical resonator that was used in the experiment under consideration is 80 cm; the diameter of its entrance opening is 120 mm; the diameter of its exit opening is 7–8 mm. The length of the smallest resonator is 21 cm; the diameter of its entrance opening is 60 mm, and of the exit opening 7–8 mm.

If the experiments are demonstrated in a large lecture hall, then the paper pinwheels should be projected in shadow projection onto a screen.

4. A ball in a jet of gas or liquid. In the steady flow of a liquid or gas through a tube of variable cross section, the pressure inside the jet changes: where the cross section of the tube is narrower, the pressure is lower, and conversely. Consequently, at the place where the cross section of the tube changes abruptly, a pressure difference arises.

Let us consider a lecture experiment that clearly demonstrates the occurrence of such a pressure difference.

A glass tube, one half of which has a cross section approximately 16 times greater than the other, is connected by means of a rubber tube to a cylinder containing liquid carbon dioxide under high pressure. Into the wide part of the tube

a ball (celluloid, ebonite, or made of duralumin) is lowered, its diameter being 1–1.5 mm smaller than the internal diameter of the wide part of the tube (Fig. 8, a).

The tube of variable cross-section is first placed vertically, together with the ball lowered into it, so that the wide part of the tube is at the top. Then the valve of the reducer to which the balloon is connected is slowly opened, and a fairly strong jet of gas, usually directed upward, is released. Owing to the pressure difference in the gas jet, the position of the ball at the constriction of the tube will be so stable that the tube together with the ball can be turned through 180°, and the ball will float in the jet, as shown in Fig. 8, b. The experiment looks especially impressive when it is carried out with a metal ball whose diameter is approximately 90 g.

Fig. 8.

The experiment can also be performed with a water jet.

In this case it is necessary to provide a strong flow of water through the tube. The easiest way to do this experiment is to take a celluloid ball from a table-tennis game. In this case the internal diameter of the wide part of the tube must be 40 mm, and that of the narrow part 10 mm. The total length of the tube of variable cross-section is about 400 mm.

5. “Sliced” ball. It is known that in tennis an oblique stroke of the racket on the ball can cause the ball to rotate. As a result, around the “sliced” ball there arises a double-sided distribution of the air streamlines.

In Fig. 9, a, the distribution of the air streamlines near a ball rotating about an axis perpendicular to the drawing is shown. In Fig. 9, b, the distribution of the air streamlines is given for the translational motion of the ball alone. In this case the air compressed by the flying ball flows around it from above and below. Finally, in Fig. 9, c, the distribution of the air streamlines around a ball is shown which, in its translational motion to the left, rotates about an axis perpendicular to the drawing. As a result, under the ball the velocity of the air will be greater, which determines

Fig. 9.

prevents the formation of a region of reduced pressure; above the ball the velocity of the air will be lower and the static pressure of the air relatively large. Therefore an additional pressure on the ball arises, directed downward.

A struck ball in flight can change its motion and acquire a new direction, which is difficult for the opponent to anticipate; the ball “deceives” the opponent.

To reproduce this phenomenon in the lecture hall, which vividly illustrates the Magnus effect, a special apparatus and a description of experiments with it are proposed. A small DC electric motor (weighing 25 g) is fastened rigidly to a massive stand in such a way that its axis is vertical. A rubber tube is fitted tightly onto the motor shaft; its length is 1–2 mm greater than that of the protruding end of the motor shaft. A celluloid ball from the game “table tennis” is placed on the end of the rubber tube (Fig. 10).

From above, a small disk \(a\) is placed on the ball, equipped with two cylindrical springs that press it to the ball. When the motor rotates, the celluloid ball is carried along by the rubber tube and rotates with sufficient steadiness at the motor frequency near the disk pressing it.

A strong steel spring \(b\) is attached to the common stand of the apparatus; with its help the ball can be struck both while it is rotating and when the ball is at rest. The spring is drawn back and released on impact by a trigger mechanism \(v\), which ensures, at each release of the spring, an impact on the celluloid ball that is the same in magnitude and direction.

The experiment is carried out in the following sequence. First, the celluloid ball is struck several times in succession with the motor not rotating, and the audience is convinced that the ball always flies in one direction for approximately 4–5 m.

Fig. 10.

On the demonstration table one may place some target (a small shield or box) into which the struck ball falls. If, however, the ball is struck when the motor is rotating at high speed, the ball will fly in another direction and will not hit the target.

When the motor (and consequently the ball itself) rotates clockwise, if one looks at the ball from above, the ball will be deflected by the additional pressure by 1.5–2 m to the right of the target; when the motor rotates counterclockwise, by the same distance to the left.

A few practical instructions.

1) In the circuit for switching on the motor, provision should be made for: a) a current switch for changing the rotation of the motor, and b) a rheostat for smoothly changing the current in the motor winding. It should be borne in mind that at certain motor speeds the rotation of the celluloid ball is least stable; under these conditions it should not be struck.

2) A small wooden disk, painted in different colors, should be fastened to the motor shaft. With the aid of this disk the audience determines, at a low number of motor revolutions, the direction of rotation of the ball.