SU81916A1 - Study Guide for Mathematics - Google Patents

Description

LEARNING FOCUS ON MATHEMATICS

It was filed on May 18, 1949 for jY 397396 n of the State Technical Committee of the USSR

The invention relates to educational tools that are intended to demonstrate instantaneous values of sinusoidal functions, such as projections of vectors onto a fixed axis or onto a time line.

A distinctive feature of the proposed tool is that it is made in the form of one or more graduated circles with arrow-vectors located along their diameters, one end hinged at the center of a twice larger graduated circle with a graded rotary ruler representing a fixed axis or a line of time.

The manual can be used to provide an overview of the relevant sections of mathematics and electrical engineering.

FIG. 1 shows a scheme for constructing a projection of constant in magnitude and direction of a vector on a rotating time line; in fig. 2 is a view of a training aid for demonstrating the projection of a single vector onto a non-moving axis; in fig. 3 - the same manual, but to demonstrate the projections of two vectors.

Instantaneous Sinusoidal Value

((o / -f pho) (1)

may, as is known, be represented as a projection of a vector a rotating at a speed a) on a fixed axis and as a projection of a vector a constant in magnitude and direction on a time line pn rotating at a speed s (Fig. 1).

The directions of rotation are opposite in the first and second cases.

It is also known that the geometrical place of points A will be a circle, constructed on the vector under consideration, as on the diameter. This follows directly from the consideration of the parametric equations of motion of point A, referred to those indicated in FIG. 1 axis of coordinates x and y:

№81916

.o52f;

g / u81p2f. Excluding the parameter φ, we find

. -... (2)

The described educational aid is made in the form of a circle / with a diameter of 2, which has the appearance of a vector (Fig. 2), hinged at point 3 of pedestal 4, corresponding to the beginning of the vector, with a narrow graduated ruler 5. For any relative position of the vector, the ruler segment 3-6 inside the circle / gives the value of the function by the formula (1).

The initial phase pho is mounted on a concentric graduated circle L tightly connected to the pedestal of the device.

When the projection of the rotating vector on the fixed axis is demonstrated, the ruler o is fixed, and the circle 1 bordering vector 2 rotates with it. In this case, the function argument (1) is indicated on the graded circle 7 directly by the arrow of vector 2.

When clarifying the idea of the projections of constant in magnitude and direction of the vector on the time line, the circle / with vector 2 is fixed, and the ruler 5 is rotated in the opposite direction, and the argument is measured on the scale of the circle 7 using the indicator 8 fixed to the ruler.

The manual makes it possible to demonstrate the instantaneous value of a function by the formula (1) for any valid argument values.

For a quantitative assessment, a ruler from O to a can be plotted on the ruler 5 in the 5th side of the axis of rotation 3, with a plus sign attributed to one direction, to another minus. This allows obtaining not only the magnitude of the projection, but also its sign.

To demonstrate at the same time the projections of several vectors, as well as to illustrate the phenomenon of advancing or lagging in phase, the manual depicted in FIG. 3. In this manual there are two circles with vectors, however, their number can be increased.

Subject invention

An educational mathematics tutorial, different for demonstrating instantaneous values of sinusoidal functions, for example, projection of vectors onto a fixed axis or timeline, is made in the form of one or more graded circles with arrow-vectors located along their diameters, one end of which is hinged in the center of a double graduated circle with a graduated rotating line representing a fixed axis or time line.

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Fig

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