Solution to problem 21.1.2 from the collection of Kepe O.E.

21.1.2
It is necessary to find the period of free oscillations of a mechanical system specified by the differential equation of oscillations 56q + 825q = 0, where q is a generalized coordinate.
The solution to this equation is a function of the form q(t) = Asin(oht + φ), where A is the amplitude of oscillations, ω is the angular frequency of oscillations, φ is the initial phase of oscillations.
The angular frequency of oscillations is determined from the equation ω = sqrt(k/m), where k is the stiffness coefficient of the system, m is its mass.
Thus, it is necessary to find the stiffness coefficient k and the mass m of the system in order to determine the angular frequency and, therefore, the period of free oscillations.
Solving the equation 56q + 825q = 0, we get k/m = 825/56.
It follows from this that k = (825/56)m.
The period of free oscillations is determined by the formula T = 2π/ω. Substituting the expression for the angular frequency, we obtain T = 2πsqrt(m/k).
Replacing k with the expression (825/56)m, we get T = 1.64sqrt(m).
Thus, to completely solve the problem it is necessary to know the mass of the mechanical system.

Solution to problem 21.1.2 from the collection of Kepe O..

This digital product is a solution to problem 21.1.2 from the collection of Kepe O.. on theoretical mechanics.

This solution describes in detail the process of determining the period of free oscillations of a mechanical system specified by the differential equation of oscillations 56q + 825q = 0, where q is a generalized coordinate.

The design of this digital product is made in a beautiful html format, which makes the material easy to read and understand.

By purchasing this digital product, you will receive a complete and understandable solution to problem 21.1.2 from the collection of Kepe O.. on theoretical mechanics.

This digital product is a solution to problem 21.1.2 from the collection of Kepe O.?. in theoretical mechanics. To solve the problem, it is necessary to find the period of free oscillations of a mechanical system specified by the differential equation of oscillations 56q + 825q = 0, where q is a generalized coordinate. The solution to this equation is a function of the form q(t) = Asin(ωt + φ), where A is the amplitude of oscillations, ω is the angular frequency of oscillations, φ is the initial phase of oscillations. The angular frequency of oscillations is determined from the equation ω = sqrt(k/m), where k is the stiffness coefficient of the system, m is its mass.

To determine the period of free oscillations, you must first find the stiffness coefficient k and the mass m of the system using the equation 56q + 825q = 0. Solving this equation, we obtain k/m = 825/56, which implies that k = (825/56)*m .

The period of free oscillations is determined by the formula T = 2π/ω. Substituting the expression for the angular frequency, we obtain T = 2π*sqrt(m/k). Replacing k with the expression (825/56)m, we get T = 1.64sqrt(m).

The presented solution contains all the necessary formulas and step-by-step instructions for determining the period of free vibrations, including finding the stiffness coefficient and mass of the system. The design of this digital product is made in a beautiful html format, which makes the material easy to read and understand. By purchasing this digital product, you will receive a complete and understandable solution to problem 21.1.2 from the collection of Kepe O.?. on theoretical mechanics, as well as a convenient format for studying the material. The answer to the problem is 1.64.

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For problem 21.1.2 from the collection of Kepe O.?. it is required to solve the differential equation of oscillations of a mechanical system, given in the form 56q + 825q = 0, where q is a generalized coordinate.

To find the free oscillation period of a mechanical system, you first need to find a solution to the differential equation. To do this, you can solve the characteristic equation, which is obtained by replacing q(t) with exp(rt) and then solving the resulting equation.

The characteristic equation for this differential equation is 56r^2 + 825 = 0.

Having solved it, we get two roots: r1 = isqrt(825/56) and r2 = -isqrt(825/56).

Since the roots are complex conjugate and have an imaginary part, the solution to the differential equation can be represented as q(t) = e^0.5t(Acos(wt) + Bsin(wt)), where A and B are arbitrary constants, and w = sqrt(825/56) is the oscillation frequency.

The period of free oscillations is defined as T = 2*pi/w. Substituting the value of w into the formula, we get T = 1.64 (rounded to the nearest hundredth).

Thus, the answer to problem 21.1.2 from the collection of Kepe O.?. equals 1.64.

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