Question

Материальная точка массой m = 2 г совершает гармонические колебания. В некоторый момент времени смещение точки x = 5 см, скорость v = 20 см/с, ускорение a = 80 см/с2. Найдите круговую частоту 0, период Т, фазу колебания в заданный момент времени, а также амплитуду колебаний xm и полную энергию Е колеблющейся точки.

Answer 1

Ответ:

Ответ : круговая частота = 4 рад/сек, период = π/2 = 1,57 сек, фаза колебаний в заданный момент времени = 45°, амплитуда = 7,07 (5·√2) см, полная энергия 80 мкДж.

Answer 2

Given Information

We are given the following information about a harmonic oscillator: - Mass of the point: m = 2 g - Displacement of the point: x = 5 cm - Velocity of the point: v = 20 cm/s - Acceleration of the point: a = 80 cm/s^2

We need to find the following: - Angular frequency: ω - Period: T - Phase of oscillation at the given moment - Amplitude of oscillation: xm - Total energy of the oscillating point: E

Calculating Angular Frequency (ω)

The angular frequency (ω) of a harmonic oscillator can be calculated using the formula:

ω = √(k/m)

where k is the spring constant and m is the mass of the point.

Since the spring constant is not given, we need to find it using the given information. The spring constant can be calculated using the formula:

k = m * a

Substituting the given values, we have:

k = 2 g * 80 cm/s^2

Converting the mass to kg and the acceleration to m/s^2:

k = 0.002 kg * 0.8 m/s^2

Now we can calculate the angular frequency:

ω = √(0.002 kg * 0.8 m/s^2 / 0.002 kg) = √(0.8 m/s^2) = 0.8944 rad/s.

Calculating Period (T)

The period (T) of a harmonic oscillator can be calculated using the formula:

T = 2π/ω

Substituting the value of ω we calculated earlier:

T = 2π/0.8944 rad/s = 7.024 s.

Calculating Phase of Oscillation

The phase of oscillation at a given moment can be calculated using the formula:

φ = arctan(v/(ωx))

Substituting the given values:

φ = arctan(20 cm/s / (0.8944 rad/s * 5 cm))

Converting the velocity and displacement to m/s and m respectively:

φ = arctan(0.2 m/s / (0.8944 rad/s * 0.05 m))

Now we can calculate the phase:

φ ≈ arctan(4.472) ≈ 1.329 rad.

Calculating Amplitude of Oscillation (xm)

The amplitude of oscillation (xm) can be calculated using the formula:

xm = √(x^2 + (v/ω)^2)

Substituting the given values:

xm = √((5 cm)^2 + (20 cm/s / 0.8944 rad/s)^2)

Converting the displacement and velocity to m and m/s respectively:

xm = √((0.05 m)^2 + (0.2 m/s / 0.8944 rad/s)^2)

Now we can calculate the amplitude:

xm ≈ √(0.0025 + 0.0448) ≈ 0.215 m.

Calculating Total Energy of the Oscillating Point (E)

The total energy of the oscillating point can be calculated using the formula:

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

Substituting the values of k and xm we calculated earlier:

E = (1/2) * (0.002 kg * 0.8 m/s^2) * (0.215 m)^2

Now we can calculate the total energy:

E ≈ 4.36 * 10^-5 J.

Summary

Based on the given information, we have calculated the following: - Angular frequency: ω = 0.8944 rad/s - Period: T = 7.024 s - Phase of oscillation at the given moment: φ ≈ 1.329 rad - Amplitude of oscillation: xm ≈ 0.215 m - Total energy of the oscillating point: E ≈ 4.36 * 10^-5 J.

Please note that these calculations assume ideal conditions for a simple harmonic oscillator and may not account for any external factors or damping effects.