Question

Тіло масою 2кг виконує коливання на пружині жорсткість якої 200Н/м з амплітудою 8см. Знайти період, частоту і циклічну частоту коливань. Запишіть рівняння х=х(t) гармонічних коливань тіла .

Answer 1

Ответ:

Объяснение:

Дано:

m = 2 кг

k = 200 Н/м

A = Δx = 8 см = 0,08 м

______________

T - ?

ν - ?

ω - ?

x(t) - ?

Период колебаний пружинного маятника:

T = 2π·√ (m/k) = 2·3,14·√ (2 / 200)  ≈  0,63 с

Частота:

ν = 1 / T = 1 / 0,63 ≈ 1,6 Гц

Циклическая частота:

ω = 2π / T = 2·3,14 / 0,63 ≈ 10 c⁻¹

Уравнение колебаний:

x(t) = A·sin (ω·t)

x(t) = 0,08·sin (10·t)  м

Answer 2

Calculation of Period, Frequency, and Angular Frequency

To find the period, frequency, and angular frequency of the oscillations, we can use the following formulas:

Period (T): The time taken for one complete oscillation. Frequency (f): The number of oscillations per unit time. Angular Frequency (ω): The rate of change of angle with respect to time.

The formulas are related as follows: T = 1/f ω = 2πf

Given: Mass of the body (m) = 2 kg Stiffness of the spring (k) = 200 N/m Amplitude (A) = 8 cm = 0.08 m

To find the period, we can use the formula: T = 2π√(m/k) [[1]]

Substituting the given values: T = 2π√(2/200) = 2π√(1/100) = 2π/10 = π/5 ≈ 0.628 s

To find the frequency, we can use the formula: f = 1/T

Substituting the value of T: f = 1/(π/5) = 5/π ≈ 1.591 Hz

To find the angular frequency, we can use the formula: ω = 2πf

Substituting the value of f: ω = 2π(1.591) ≈ 10π ≈ 31.42 rad/s

Equation of Harmonic Oscillations

The equation of harmonic oscillations for the body can be written as: x = A cos(ωt + φ)

Where: x = displacement of the body from its mean position A = amplitude of the oscillations ω = angular frequency of the oscillations t = time φ = phase constant

In this case, the given amplitude is 0.08 m and the angular frequency is 31.42 rad/s.

Therefore, the equation of harmonic oscillations for the body is: x = 0.08 cos(31.42t + φ)

Please note that the phase constant (φ) is not provided in the given information, so it cannot be determined without additional details.

I hope this helps! Let me know if you have any further questions.