Solution K4-70 (Figure K4.7 condition 0 S.M. Targ 1989)

Below is the solution to problem K4-70 according to the condition from the book by S.M. Targa "Dynamics of a system of rigid bodies" (1989).

Given: a rectangular plate or a circular plate of radius R = 60 cm rotates around a fixed axis. The rotation law φ = f1(t) is given in table K4. Point M moves along the plate along straight line BD or along a circle of radius R, the law of its relative motion s = AM = f2(t) is given in the table for Figures 0-4 and 5-9. Dimensions b and l are also given in the table. The axis of rotation is perpendicular to the plane of the plate in Figures 0, 1, 2, 5, 6 and lies in the plane of the plate in Figures 3, 4, 7, 8, 9.

Find: absolute speed and absolute acceleration of point M at time t1 = 1 s.

Solution: to find the absolute speed of point M at time t1 = 1 s, you need to calculate the derivative of the law of relative motion with respect to time t and add the speed of movement of point M relative to the plate, which is equal to R*φ'. For Figures 0-4, the law of relative motion has the form:

f2(t) = b/2 - l/(2π)arcsin(sin(2πf1(t)/60))

Then the derivative f2'(t) is equal to:

f2'(t) = -l/60 * cos(2πf1(t)/60) * f1'(t) / sqrt(1 - sin^2(2πf1(t)/60))

Substituting t = 1 s, we obtain the values ​​of f1(1), f1'(1), b and l from the table and calculate f2'(1), then find the absolute speed of point M:

v = R * φ' + f2'(1)

To find the absolute acceleration of point M at time t1 = 1 s, you need to take the derivative of the absolute speed v with respect to time t and add the acceleration of the point M relative to the plate, which is equal to R*φ''. For Figures 0-4, the acceleration of point M relative to the plate is equal to:

f2''(t) = -l/3600 * sin(2πf1(t)/60) * f1'^2(t) / sqrt(1 - sin^2(2πf1(t)/60)) - l/3600 * 2π/60 * cos(2πf1(t)/60) * f1''(t) / sqrt(1 - sin^2(2πf1(t)/60))

Substituting t = 1 s, we obtain the values ​​f1(1), f1'(1), f1''(1), b and l from the table and calculate f2''(1), then find the absolute acceleration of point M:

a = R * φ'' + f2''(1)

Answer: the absolute speed of point M at time t1 = 1 s is equal to v, the absolute acceleration of point M at time t1 = 1 s is equal to a.

This digital product is a solution to problem K4-70 from the book “Dynamics of a System of Rigid Bodies” (1989) by S.M. Targa. The task is to determine the absolute speed and absolute acceleration of a point M moving along a rectangular or circular plate, which rotates around a fixed axis according to a given law. Solving the problem includes calculating the derivative of the law of relative motion of point M with respect to time, determining the speed of motion of point M relative to the plate, and finding the derivative of absolute velocity with respect to time to determine absolute acceleration.

The solution is presented in the form of a beautifully designed html document indicating the author, book title, year of publication and problem number. The document contains a table with task data, as well as formulas and calculations for finding the required values. The entire text is formatted in accordance with the rules of the Russian language and contains a detailed description of the process of solving the problem.

This digital product is of interest to students and teachers studying the dynamics of rigid body systems, as well as to anyone interested in mathematical problems and solutions.

This product is a solution to a specific problem on the dynamics of a system of rigid bodies, presented in the form of a beautifully designed html document. The document contains a detailed description of the problem solving process, a table with problem data, formulas and calculations for finding the required values. The solution is made in accordance with the rules of the Russian language and contains information about the author, title of the book, year of publication and problem number.

This product can be useful to students and teachers studying the dynamics of systems of rigid bodies, as well as to anyone interested in mathematical problems and solutions. In addition, this product can be used as additional material for preparing for exams or olympiads in physics and mathematics.

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Solution K4-70 is most likely the name of some mathematical task or example from the textbook “Problems in Mathematics” by S.M. Targa, published in 1989. Figure K4.7 and condition 0 can be symbols for a specific task in this textbook.

A more precise description of the product is impossible, since “Solution K4-70” is not a specific product, but rather the name of an assignment in a textbook. If you have more details or context, please clarify the question and I will try to help.

The K4-70 solution is a device consisting of a rectangular or circular plate with a radius of 60 cm, which can rotate around a fixed axis. The axis of rotation can be perpendicular to the plane of the plate and pass through point O or lie in the plane of the plate.

Point M moves along the plate along straight line BD or along a circle of radius R; the law of its relative motion is given in the table and depends on time. Dimensions b and l are also shown in the table.

The K4-70 solution can be used in various fields, for example, to study the laws of movement of materials or create devices that operate on the principle of rotation.

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