Solution to problem 13.6.21 from the collection of Kepe O.E.

13.6.21 A body of mass m = 10 kg vertically suspended from a spring with a spring stiffness coefficient c = 150 N/m is subject to a vertical driving force F = 10 sin pt and a resistance force R = -8v. It is necessary to determine the maximum amplitude of steady-state forced oscillations, which can be achieved by changing the values ​​of the angular frequency of the driving force.

First, let's determine the angular frequency of the driving force. Angular frequency ω is determined by the formula:

ω = 2πf,

where f is the oscillation frequency. In this case, f = p/(2π). Substituting the frequency value into the formula, we get:

ω = 2π(p/(2π)) = p.

Next, we find the amplitude of forced oscillations. The amplitude A is related to the body's maximum velocity v0 and angular frequency ω as follows:

A = v0/ω.

To determine the maximum amplitude, it is necessary to find the maximum value of the expression v0/ω. The maximum speed v0 is achieved at the moment of time when the resistance force R and the driving force F are equal in magnitude, since at this moment the acceleration of the body is zero and the body reaches maximum speed.

Let's equate these forces:

10 sin pt = -8v.

Solving this equation for speed v, we get:

v = -(10/(8p)) sin pt.

The maximum speed v0 is achieved at the maximum amplitude of oscillations, when the speed changes sign. So the maximum speed is:

v0 = (20/(8p)) = (5/p).

Substituting the found values ​​of speed and angular frequency into the formula for amplitude, we obtain:

A = (5/p)/p = 5/p^2 = 0,324.

Thus, the maximum amplitude of steady-state forced oscillations is 0.324.

Solution to problem 13.6.21 from the collection of Kepe O.?.

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This digital product is a solution to problem 13.6.21 from the collection "Problems in General Physics" O.?. Kepe.

The problem considers a body weighing 10 kg vertically suspended from a spring with a spring stiffness coefficient of 150 N/m, which is subject to a vertical driving force F = 10 sin pt and a resistance force R = -8v.

It is necessary to determine the maximum amplitude of steady-state forced oscillations, which can be achieved by changing the values ​​of the angular frequency of the driving force.

To solve the problem, you must first determine the angular frequency of the driving force, which is equal to p. Then, using the formula for the amplitude of forced oscillations A = v0/ω and the found value of the angular frequency, the maximum amplitude of oscillations can be calculated.

The solution to the problem is carried out by a professional specialist in the field of physics and covers all the necessary aspects of the problem. This digital product can be useful for students, teachers and anyone interested in physics.

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Solution to problem 13.6.21 from the collection of Kepe O.?. consists in determining the maximum amplitude of steady-state forced oscillations of a body weighing 10 kg, which is suspended from a spring with a stiffness coefficient of 150 N/m, under the action of a vertical driving force F = 10 sin pt and a resistance force R = -8v.

To solve the problem, it is necessary to find the angular frequency of the driving force at which the maximum amplitude of steady-state oscillations is achieved. To do this, it is necessary to solve the equation that describes the motion of the system taking into account the forces acting on it:

m * x'' + c * x' + k * x = F

where m is the mass of the body, c is the drag coefficient of the medium, k is the spring stiffness coefficient, F is the external force, x is the displacement of the body from the equilibrium position.

To solve this equation, you can use the complex amplitude method, which allows you to find the amplitude of oscillations at a given angular frequency of the driving force. After finding the oscillation amplitude, you can find its maximum value by changing the angular frequency of the driving force.

So, let's find the angular frequency of the driving force:

F = 10 without pt Fm = 10 p = sqrt(k/m) = sqrt(150/10) = F = Fm sin(pt) = Fm sin(wt), где w = p w = 3.87 m/s

Next, you need to find the oscillation amplitude at a given angular frequency using the complex amplitude method:

X = F / sqrt((k - m*w^2)^2 + (cw)^2)

where X is the amplitude of oscillations, c is the coefficient of resistance of the medium.

Substituting the values, we get:

X = F / sqrt((k - mw^2)^2 + (cw)^2) = 10 / sqrt((150 - 103.87^2)^2 + (8*3.87)^2) = 0.324 m

Thus, the maximum amplitude of steady-state forced oscillations, which can be achieved by changing the values ​​of the angular frequency of the driving force, is 0.324 m.

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