Solution to problem 20.5.8 from the collection of Kepe O.E.

20.5.8 Kinetic energy of a mechanical system T = 2x2, potential energy P = 4x. It is necessary to determine the generalized speed of the system x at time t = 3 s, if x|t = 0 = 13 m/s. (Answer 10).

To solve the problem, it is necessary to use the law of conservation of energy, which states that the sum of the kinetic and potential energies of the system remains constant. From the conditions of the problem, the values ​​of kinetic and potential energies at time t = 0 s are known.

Kinetic energy T = 2x2, potential energy P = 4x.

Therefore, the total energy of the system at the initial moment of time is equal to:

E = T + P = 2x2 + 4x = 2x(x + 2).

From the law of conservation of energy it follows that the total energy of the system at any time remains constant. Thus:

E = 2x(x + 2) = const.

To determine the generalized speed of the system x at time t = 3 s, it is necessary to use the equation of motion of the system:

x = x|t=0 + v|t=0*t + (a/2)*t^2,

where x|t=0 is the initial value of the generalized coordinate, v|t=0 is the initial value of the generalized velocity, a is the acceleration.

Dividing this equation by t^2 and taking the time derivative, we get:

a = 2(x - x|t=0 - v|t=0*t)/t^2.

Considering that the potential energy of the system is P = 4x, we can write the equation of motion in the form:

T + П = const => 2х^2 + 4х = const.

Differentiating this equation with respect to time, we obtain:

2x*v + 4v = 0.

Substituting the values ​​from the problem conditions, we get:

213v + 4v = 0,

whence v = -5.2.

Thus, the generalized speed of the system x at time t = 3 s is equal to -5.2 m/s.

Solution to problem 20.5.8 from the collection of Kepe O.?.

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The product is the solution to problem 20.5.8 from the collection of Kepe O.?.

In this problem, we are given the kinetic and potential energies of a mechanical system depending on the generalized coordinate x: T = 2x^2 and P = 4x. It is necessary to determine the generalized speed of the system x at time t=3 s, if x|t=0=13 m/s.

To solve the problem, it is necessary to use the Lagrange equation of the second kind: d/dt(dL/dx_dot) - dL/dx = 0, where L is the Lagrangian of the system.

We calculate the Lagrangian of the system: L = T - П = 2x^2 - 4x.

Next, we calculate the derivatives: dL/dx = 4x - 4 and dL/dx_dot = 4x_dot.

We substitute the second kind into the Lagrange equation and obtain the equation of motion of the system: d/dt(4x_dot) - (4x - 4) = 0.

We solve this equation and find the generalized speed of the system x at time t=3 s, if x|t=0=13 m/s: x_dot = 10 m/s. Answer: 10.

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