Question

За 100 с система совершает 100 колебаний. За тоже время амплитуда уменьшается в 2,718 раз. Найти коэффициент затухания колебаний и логарифмический декремент затухания.

Answer 1

Ответ:

Объяснение:

t=100 c    N=100    A1/A2=2.718    δ=?    Θ=?

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δ=(1/t)*ln(A1/A2)=(1/100)*ln2.718≈0.01 c⁻¹

T=t/N=100/100=1 c

Θ=δ*T=0.01*1=0.01

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Answer 2

Calculation of Damping Coefficient and Logarithmic Decrement

To find the damping coefficient and logarithmic decrement of the oscillations, we need to use the given information:

- The system completes 100 oscillations in 100 seconds. - During this time, the amplitude decreases by a factor of 2.718.

Let's denote the initial amplitude as A0 and the final amplitude after 100 oscillations as A100.

The damping coefficient (ζ) can be calculated using the formula:

ζ = (1/n) * ln(A0/A100)

where n is the number of oscillations.

In this case, n = 100, A0 = 100, and A100 = 100/2.718.

Substituting these values into the formula, we get:

ζ = (1/100) * ln(100/(100/2.718))

To find the logarithmic decrement (δ), we can use the formula:

δ = ln(A0/A1)

where A1 is the amplitude after one complete oscillation.

In this case, A1 = A0/2.718.

Substituting the values into the formula, we get:

δ = ln(100/(100/2.718))

Now, let's calculate the damping coefficient and logarithmic decrement using the given formulas.

Calculation:

Damping coefficient (ζ): ζ = (1/100) * ln(100/(100/2.718))

Logarithmic decrement (δ): δ = ln(100/(100/2.718))

Using the given formulas, we can calculate the values of ζ and δ.

Please note that the exact values cannot be determined without the specific values of A0 and A100. However, the formulas provided above can be used to calculate the damping coefficient and logarithmic decrement once the specific values are known.