Question

груз массой 400 г совершает колебания на пружине с жесткостью 250 Н/м. Амплитуда колебаний 15 см. Найдите полную механическую энергию и частоту колебаний груза. Пожалуйста, решите задание​

Answer 1

Ответ:

m = 400 г = 0,4 кг; К = 250 Н/м; А = 15 см = 0,15 м;

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Answer 2

Problem Analysis

We are given the mass of a object (400 g) and the stiffness of a spring (250 N/m). We are asked to find the total mechanical energy and the frequency of the oscillations.

Solution

To find the total mechanical energy of the oscillating mass, we need to consider both the kinetic energy and the potential energy.

The potential energy of a mass-spring system is given by the equation:

Potential Energy = (1/2) * k * x^2

where k is the stiffness of the spring and x is the displacement from the equilibrium position.

The kinetic energy of a mass-spring system is given by the equation:

Kinetic Energy = (1/2) * m * v^2

where m is the mass of the object and v is the velocity of the object.

Since the object is oscillating, the total mechanical energy is constant and equal to the sum of the potential energy and the kinetic energy.

To find the frequency of the oscillations, we can use the equation:

Frequency = (1 / (2 * pi)) * sqrt(k / m)

where k is the stiffness of the spring and m is the mass of the object.

Let's calculate the total mechanical energy and the frequency of the oscillations.

Calculation

Given: - Mass of the object (m) = 400 g = 0.4 kg - Stiffness of the spring (k) = 250 N/m - Amplitude of the oscillations (A) = 15 cm = 0.15 m

To find the total mechanical energy, we need to find the maximum displacement (x) from the equilibrium position. The maximum displacement is equal to the amplitude of the oscillations.

Maximum Displacement (x) = Amplitude (A) = 0.15 m

Now we can calculate the potential energy:

Potential Energy = (1/2) * k * x^2

Potential Energy = (1/2) * 250 N/m * (0.15 m)^2

Potential Energy = 1.875 J

Next, we can calculate the kinetic energy. At the maximum displacement, the velocity is zero, so the kinetic energy is also zero.

Kinetic Energy = 0 J

The total mechanical energy is the sum of the potential energy and the kinetic energy:

Total Mechanical Energy = Potential Energy + Kinetic Energy

Total Mechanical Energy = 1.875 J + 0 J

Total Mechanical Energy = 1.875 J

Finally, we can calculate the frequency of the oscillations:

Frequency = (1 / (2 * pi)) * sqrt(k / m)

Frequency = (1 / (2 * pi)) * sqrt(250 N/m / 0.4 kg)

Frequency ≈ 3.98 Hz

Answer

The total mechanical energy of the oscillating mass is approximately 1.875 J, and the frequency of the oscillations is approximately 3.98 Hz.

Please note that the calculations are approximate due to rounding off the values.