Final answer:
To solve for the damping constant c and the spring constant k, we can use the given information that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. The damping constant can be calculated using the natural logarithm of the amplitude ratio, and the spring constant can be calculated using the frequency formula for a damped harmonic oscillator.
Step-by-step explanation:
To solve for the damping constant c and the spring constant k, we can use the given information that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. In a damped harmonic motion, the amplitude decreases exponentially as the number of cycles increases.
The amplitude of the 30th cycle can be calculated using the formula:
An = A1 * e(-cn/2m)
where An is the amplitude of the nth cycle, A1 is the amplitude of the 1st cycle, n is the number of cycles, c is the damping constant, and m is the mass.
Based on the given information, we have:
A30 = 0.2 * A1
Substituting the values, we get:
0.2 * A1 = A1 * e(-30c/2m)
Calculating the damping constant c:
To solve for c, we need to rearrange the equation:
e(-30c/2m) = 0.2
Taking the natural logarithm of both sides:
(-30c/2m) = ln(0.2)
Simplifying the equation:
c = (2m/30) * ln(0.2)
Calculating the spring constant k:
Given the formula for the frequency of a damped harmonic oscillator:
f = (1 / 2π) * sqrt(k/m - c²/4m²)
We can rearrange the formula to solve for k:
k = (4π²m * f²) + (2cπmf)
Substituting the values:
k = (4π² * 100) + (2 * c * π * 100)
where f is the frequency, m is the mass, c is the damping constant.