Solution to problem 13.5.19 from the collection of Kepe O.E.

13.5.19

For a material point, oscillatory motion is described by the differential equation x + 6x' + 50x = 0. It is necessary to determine the period of damped oscillations.

Answer: 0.981

Let's consider this differential equation of oscillatory motion of a material point and find a solution. The characteristic equation has the form:

r^2 + 6r + 50 = 0

The discriminant of this equation is:

D = b^2 - 4ac = 6^2 - 4150 = -116

Since the discriminant is negative, the roots of the equation will be complex:

r1 = -3 + 4i r2 = -3 - 4i

Thus, the general solution to the differential equation has the form:

x(t) = e^(-3t)(с1cos(4t) + с2sin(4t))

where c1 and c2 are arbitrary constants.

The period of damped oscillations is determined by the formula:

T = 2*pi/h,

where ω is the natural frequency of oscillations, equal to 4.

Then the period of damped oscillations will be equal to:

T = 2*pi/4 = pi/2 ≈ 0.981.

Solution to problem 13.5.19 from the collection of Kepe O.?.

This digital product is the solution to problem 13.5.19 from the collection of Kepe O.?. in physics. The solution is presented in the form of a detailed description of the solution steps using differential equations and mathematical formulas.

This solution will be useful for students and physics teachers who study oscillatory processes in mechanics. It will help you better understand the theoretical foundations of oscillations and consolidate the material in practice.

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Buy this solution and improve your physics knowledge today!

This product is a solution to problem 13.5.19 from the collection of Kepe O.?. in physics. The problem is to determine the period of damped oscillations of a material point for which a differential equation of oscillatory motion is given. The solution is presented in the form of detailed mathematical calculations and solution steps, including finding the characteristic equation, determining the roots of this equation, and deriving the general solution to the differential equation. The solution then uses a formula to determine the period of damped oscillations, which depends on the natural frequency of oscillations. The result is an answer in the form of a number equal to the period of damped oscillations of the material point.

This product will be useful to students and physics teachers who study oscillatory processes in mechanics. The solution will help you better understand the theoretical foundations of oscillations and consolidate the material in practice. It also has a convenient format and beautiful design in html, which makes its use more comfortable and enjoyable. The solution can be purchased to improve your knowledge in physics today.

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Problem 13.5.19 from the collection of Kepe O.?. consists in determining the period of damped oscillations of a material point moving in accordance with the differential equation x + 6x + 50x = 0. To solve this problem it is necessary to find a general solution to the differential equation and determine the values ​​of the constants using the initial conditions. Then, using the found values ​​of the constants, the period of damped oscillations can be calculated.

Answer to problem 13.5.19 from the collection of Kepe O.?. is 0.981.

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