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?
To find the answers to these questions, we can use the equation for damped harmonic motion:
mx'' + bx' + kx = 0
This equation describes how the mass, spring constant, damping constant, and displacement of the system are related. It's a second-order differential equation, which means we need to solve it to find the behavior of the system over time.
Let's solve each question step by step:
(a) To find out how long it takes for the amplitude of the oscillations to decrease to 1% of its original value, we need to determine the decay time or the time it takes for the amplitude to decrease to approximately 0.01 times its initial value.
1. For a damped harmonic oscillator, the decay time is related to the damping constant by the formula:
τ = 2π / ωd
where τ is the decay time and ωd is the damped angular frequency.
2. The damped angular frequency can be found using the formula:
ωd = √(ω0^2 - (b/2m)^2)
where ω0 is the natural angular frequency.
3. The natural angular frequency is given by:
ω0 = √(k/m)
where k is the spring constant and m is the mass.
4. Finally, we can substitute ωd into the decay time formula to find the answer.
(b) To find the damping constant needed to reduce the amplitude of the oscillations by 70% in 3 seconds, we need to determine the damping constant that satisfies this condition.
1. Firstly, we know that the amplitude of a damped harmonic oscillator at any given time t is given by:
A(t) = A0 * e^(-bt/2m)
where A(t) is the amplitude at time t, A0 is the initial amplitude, b is the damping constant, and m is the mass.
2. Since we want the amplitude to be reduced by 70%, we can use the formula:
0.3A0 = A0 * e^(-3b/2m)
3. Solving this equation for b will give us the desired damping constant.
Now, let's calculate the answers using the given values:
(a) For a 3-kg mass, k = 142 N/m, and b = 14 kg/s,
1. Calculate the natural angular frequency:
ω0 = √(142 N/m / 3 kg) ≈ 6.12 rad/s
2. Calculate the damped angular frequency:
ωd = √((6.12 rad/s)^2 - ((14 kg/s) / (2 * 3 kg))^2) ≈ 6.08 rad/s
3. Calculate the decay time:
τ = 2π / ωd = 2π / 6.08 rad/s ≈ 1.03 s
Therefore, it will take approximately 1.03 seconds for the amplitude of the oscillations to decrease to 1% of its original value.
(b) For reducing the amplitude by 70% in 3 seconds,
1. Using the amplitude equation and substituting the values:
0.3A0 = A0 * e^(-3b / (2 * 3 kg))
0.3 = e^(-b / 2 kg/s)
2. Solve for b:
-ln(0.3) = -b / 2 kg/s
b ≈ 1.02 kg/s
Therefore, the damping constant should be approximately 1.02 kg/s to reduce the amplitude of the oscillations by 70% in 3 seconds.