A 3-kg mass attached to a spring with k = 142 N/m is oscillating in a vat of oil, which damps the oscillations.
(a) If the damping constant of the oil is b = 14 kg/s, how long will it take the amplitude of the oscillations to decrease to 1% of its original value?
(b) What should the damping constant be to reduce the amplitude of the oscillations by 70% in 3 s?
(a) To find the time it takes for the amplitude of the oscillations to decrease to 1% of its original value, we need to determine the damping ratio of the system.
The damping ratio (ζ) is given by the formula:
ζ = b / (2 * √(m * k))
where b is the damping constant, m is the mass, and k is the spring constant.
In this case, m = 3 kg, k = 142 N/m, and b = 14 kg/s.
ζ = 14 / (2 * √(3 * 142))
Simplifying:
ζ = 0.123
The time it takes for the amplitude to decrease to 1% of its original value (t₁) can be found using the formula:
t₁ = (-ln(0.01)) / (ζ * ω₀)
where ω₀ is the natural frequency of the system and can be calculated using the formula:
ω₀ = √(k / m)
ω₀ = √(142 / 3)
Simplifying:
ω₀ ≈ 8.22 rad/s
t₁ = (-ln(0.01)) / (0.123 * 8.22)
Using a calculator, we find:
t₁ ≈ 19.5 seconds
So, it will take approximately 19.5 seconds for the amplitude of the oscillations to decrease to 1% of its original value.
(b) To find the damping constant required to reduce the amplitude of the oscillations by 70% in 3 seconds, we need to determine the damping ratio for this scenario.
Using the formula for damping ratio:
ζ = (-ln(0.7)) / (ω₀ * t)
We know that ω₀ = √(k / m) as calculated earlier, and t = 3 seconds.
ζ = (-ln(0.7)) / (√(142 / 3) * 3)
Using a calculator, we find:
ζ ≈ 0.196
Now, to find the damping constant (b₂) required for this damping ratio, we use the formula:
b₂ = 2 * ζ * √(m * k)
b₂ = 2 * 0.196 * √(3 * 142)
Simplifying:
b₂ ≈ 8.89 kg/s
So, the damping constant should be approximately 8.89 kg/s to reduce the amplitude of the oscillations by 70% in 3 seconds.
To answer both questions, we need to use the equation of damped harmonic motion:
mx'' + bx' + kx = 0,
where m is the mass, b is the damping constant, k is the spring constant, x(t) is the displacement of the mass as a function of time t, and x'' and x' denote the first and second derivatives of x with respect to t, respectively.
(a) To find the time it takes for the amplitude to decrease to 1% of its original value, we need to find the decay time constant (τ) first. The decay time constant is given by the formula:
τ = m / b.
Therefore, τ = 3 kg / 14 kg/s = 0.214 s.
The equation for the amplitude of a damped harmonic oscillator is given by:
A(t) = A(0) * e^(-t/τ),
where A(t) is the amplitude at time t, A(0) is the initial amplitude, and e is the base of the natural logarithm.
We know that we want A(t) to be 1% of A(0). Let's call A(t) = 0.01A(0).
0.01A(0) = A(0) * e^(-t/τ).
Dividing both sides by A(0) gives:
0.01 = e^(-t/τ).
To solve for t, we can take the natural logarithm of both sides:
ln(0.01) = -t/τ.
Rearranging, we can find t:
t = -τ * ln(0.01).
Calculating this value will give us the time it takes for the amplitude to decrease to 1% of its original value.
(b) To find the damping constant (b) required to reduce the amplitude of the oscillations by 70% in 3 s, we can use the same formula as above:
A(t) = A(0) * e^(-t/τ).
We want to find the damping constant b such that A(3) = 0.3A(0), where A(t) is the amplitude at time t = 3.
0.3A(0) = A(0) * e^(-3/τ).
Dividing both sides by A(0) gives:
0.3 = e^(-3/τ).
Taking the natural logarithm of both sides:
ln(0.3) = -3/τ.
Rearranging, we can find τ:
τ = -3 / ln(0.3).
Then, using the formula τ = m / b, we can solve for the damping constant (b) by rearranging the formula:
b = m / τ.
Substituting the given values:
m = 3 kg,
τ = -3 / ln(0.3),
we can calculate b to find the required damping constant.