Question

Тело массой 1 кг колеблется на пружине с амплитудой 0,02 м. Максимальное ускорение тела равно 0,3 м/с2. Определите полную механическую энергию колебаний.

Answer 1

E=k*xm^2/2
m*am=k*xm
E=m*am*xm/2=1*0,3*0,02/2=0,003 Дж

Answer 2

Calculation of the Total Mechanical Energy of Oscillations

To calculate the total mechanical energy of the oscillations, we need to consider both the kinetic energy and the potential energy of the system.

The kinetic energy (KE) of the oscillating body can be calculated using the formula:

KE = (1/2) * m * v^2

where m is the mass of the body and v is its velocity.

The potential energy (PE) of the oscillating body can be calculated using the formula:

PE = (1/2) * k * x^2

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

The total mechanical energy (E) of the oscillations is the sum of the kinetic and potential energies:

E = KE + PE

Let's calculate the total mechanical energy using the given information.

Given: - Mass of the body (m) = 1 kg - Amplitude of oscillation (A) = 0.02 m - Maximum acceleration (a) = 0.3 m/s^2

To calculate the total mechanical energy, we need to find the velocity (v) and the spring constant (k).

Calculating the Velocity (v)

The maximum acceleration (a) is related to the amplitude (A) and the angular frequency (ω) by the equation:

a = ω^2 * A

Rearranging the equation, we can solve for ω:

ω = sqrt(a / A)

The velocity (v) can be calculated using the equation:

v = ω * A

Substituting the values, we can calculate the velocity:

v = sqrt(a / A) * A

Let's calculate the velocity:

v = sqrt(0.3 / 0.02) * 0.02

Using a calculator, we find that v ≈ 0.7746 m/s.

Calculating the Spring Constant (k)

The angular frequency (ω) can be calculated using the equation:

ω = sqrt(k / m)

Rearranging the equation, we can solve for k:

k = m * ω^2

Substituting the values, we can calculate the spring constant:

k = 1 * (sqrt(0.3 / 0.02))^2

Using a calculator, we find that k ≈ 225 N/m.

Calculating the Total Mechanical Energy (E)

Now that we have the velocity (v) and the spring constant (k), we can calculate the total mechanical energy (E) using the formulas mentioned earlier:

KE = (1/2) * m * v^2

PE = (1/2) * k * x^2

E = KE + PE

Since the body is at the maximum displacement (amplitude), the potential energy (PE) is at its maximum value. Therefore, the total mechanical energy (E) is equal to the potential energy (PE).

Let's calculate the total mechanical energy:

E = (1/2) * k * A^2

Substituting the values, we can calculate the total mechanical energy:

E = (1/2) * 225 * 0.02^2

Using a calculator, we find that E ≈ 0.045 J.

Therefore, the total mechanical energy of the oscillations is approximately 0.045 Joules.