Solution to problem 15.2.10 from the collection of Kepe O.E.

15.2.10. A load weighing 1 kg performs free vibrations according to the law x = 0.1 sin 10 t. Spring stiffness coefficient c = 100 N/m. Determine the total mechanical energy of the load at x = 0.05 m, if at x = 0 the potential energy is zero. (Answer 0.5)

To solve the problem, we need to find the kinetic energy of the load at a given value of the x coordinate and calculate the total mechanical energy, which is equal to the sum of the kinetic and potential energies.

First of all, let’s find the speed of the load at the point with coordinate x = 0.05 m. To do this, we differentiate the formula x = 0.1 sin 10 t with respect to time:

v = dx/dt = 1*cos(10t) = cos(10t) м/с

Then we find the kinetic energy of the load at x = 0.05 m:

E_To = (m*v²)/2 = (1*(cos(10t))²)/2 = 0,5*cos²(10t) J

Finally, the total mechanical energy of the load at x = 0.05 m is equal to:

E_nail = E_sweat + E_To = 0 + 0,5*cos²(10t) = 0.5 J

Answer: 0.5 J.

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The product in this case is the solution to problem 15.2.10 from the collection of Kepe O.?.

The task is to determine the total mechanical energy of the load at a certain value of its x coordinate. To solve it, you need to know the mass of the load (1 kg), the law of oscillations (x = 0.1 sin 10 t), the spring stiffness coefficient (c = 100 N/m) and the initial conditions (at x = 0, the potential energy is zero).

The solution to the problem is to calculate the kinetic and potential energy of the load at a given value of the x coordinate, and their summation to obtain the total mechanical energy of the load.

In this case, at x = 0.05 m, the total mechanical energy of the load is 0.5 J (answer to the problem).

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