Solution to problem 20.5.7 from the collection of Kepe O.E.

Task 20.5.7

Hopefully:

The kinetic potential of a mechanical system is determined by the expression L = 16x2 + 20x. Initial values: x|t=0 = 0, x|t = 0 = 2 m/s.

Find:

The value of the generalized x coordinate at time t = 3 s.

Answer:

To find the generalized coordinate x, it is necessary to solve the Euler-Lagrange equation:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$$

For this system:

$$\frac{\partial L}{\partial \dot{x}}=32x+20$$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=32\frac{dx}{dt}$$

$$\frac{\partial L}{\partial x}=32x$$

Substituting the expressions into the Euler-Lagrange equation, we obtain:

$$32\frac{dx}{dt}-32x=0$$

$$\frac{dx}{dt}=x$$

Solving the differential equation, we get:

$$x=What^{t}$$

Using the initial conditions, we find the constant C:

$$x|_{t=0}=0=What^{0}$$

$$C=0$$

Thus, the generalized x coordinate is zero at any time.

According to the given initial conditions, the system moves at zero speed and has no kinetic energy. The value of the generalized coordinate x does not change over time and is always equal to zero.

Solution to problem 20.5.7 from the collection of Kepe O.?.

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We present to your attention a digital product - a solution to problem 20.5.7 from the collection of Kepe O.?. This problem is related to determining the value of the generalized coordinate x of the mechanical system at time t = 3 s, if at the beginning of the movement x|t=0 = 0, x|t = 0 = 2 m/s.

Our solution is a digital product that can be purchased from our Digital Product Store. It is presented in HTML format, with a beautiful design and clear structure. In solving the problem, we will analyze in detail the solution methods used to solve this problem and provide a detailed algorithm that will help you solve this problem easily and quickly.

The first step in solving the problem is to write the Lagrange equation of the 2nd kind for a given mechanical system:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$$

For this system:

$$\frac{\partial L}{\partial \dot{x}}=32x+20$$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=32\frac{dx}{dt}$$

$$\frac{\partial L}{\partial x}=32x$$

Substituting the expressions into the Lagrange equation, we get:

$$32\frac{dx}{dt}-32x=0$$

Solving the differential equation, we get:

$$x=What^{t}$$

Using the initial conditions, we find the constant C:

$$x|_{t=0}=0=What^{0}$$

$$C=0$$

Thus, the generalized x coordinate is zero at any time. According to the given initial conditions, the system moves at zero speed and has no kinetic energy. The value of the generalized coordinate x does not change over time and is always equal to zero.

Purchase our solution to problem 20.5.7 from the collection of Kepe O.?. right now and get a unique opportunity to better understand the kinetic potential of a mechanical system and solve this problem without problems!

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Problem 20.5.7 from the collection of Kepe O.?. is associated with determining the value of the generalized coordinate at time t=3 seconds for a mechanical system with kinetic potential L=16x^2+20x, where x is the generalized coordinate. Initial conditions of the problem: x|t=0=0, x'|t=0=2 m/s.

To solve the problem, it is necessary to use the principle of least action, according to which the true trajectory of the system corresponds to the extremum of the action. The action for this system can be written as an integral of the Lagrange function L(x,x',t):

S = ∫L(x, x', t)dt

where L(x,x',t) = T - V - kinetic and potential energies of the system, respectively.

To find the value of the generalized coordinate x at time t=3 seconds, it is necessary to solve the Euler-Lagrange equation for the Lagrange function L(x,x',t):

(d/dt)(∂L/∂x') - ∂L/∂x = 0

Having solved this equation, we obtain a second-order differential equation that can be solved by numerical methods. As a result of solving this equation, we obtain the value of the generalized coordinate x at time t=3 seconds, equal to 8.81.

Thus, to solve problem 20.5.7 from the collection of Kepe O.?. it is necessary to apply the principle of least action and solve the Euler-Lagrange equation for the Lagrange function L(x,x',t), and then use numerical methods to find the value of the generalized coordinate x at time t=3 seconds.

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