To find the relaxation of the pendulum and the quality factor, we need to use the formulas related to damped harmonic motion.
The formula for the amplitude of a damped harmonic oscillator after a certain number of oscillations is given by:
A = A₀ * e^(-ζ * n),
where A is the amplitude after n oscillations, A₀ is the initial amplitude, ζ is the damping factor, and e is the base of the natural logarithm.
In this case, we are given that the initial amplitude A₀ is 100, and after 10 complete oscillations, the amplitude is reduced to 50. Let's substitute these values into the equation:
50 = 100 * e^(-ζ * 10).
Now, we can solve this equation to find the damping factor (ζ):
e^(-ζ * 10) = 0.5.
Taking the natural logarithm (ln) of both sides:
-ζ * 10 = ln(0.5).
ζ = -ln(0.5) / 10.
To find the relaxation of the pendulum, we use the formula:
Relaxation = ζ * T,
where T is the period of oscillation. In this case, T = 1 second.
Relaxation = (-ln(0.5) / 10) * 1.
Finally, to find the quality factor (Q), we can use the following relationship:
Q = 1 / (2ζ).
Q = 1 / (2 * (-ln(0.5) / 10)).
Calculating the values:
ζ ≈ -0.0693, Relaxation ≈ -0.0693 seconds, Q ≈ 28.885.
Therefore, the relaxation of the pendulum is approximately -0.0693 seconds, and the quality factor is approximately 28.885.