Question

A \(3.00-\mathrm{kg}\) mass attached to a spring with \(k=140 . \mathrm{N} / \mathrm{m}\) is oscillating in a vat of oil, which damps the oscillations. a) If the damping constant of the oil is \(b=10.0 \mathrm{~kg} / \mathrm{s},\) how long will it take the amplitude of the oscillations to decrease to \(1.00 \%\) of its original value? b) What should the damping constant be to reduce the amplitude of the oscillations by \(99.0 \%\) in \(1.00 \mathrm{~s} ?\)

Short answer

Answer: It takes 9.21 seconds for the amplitude to decrease to 1% of its original value. Question: What damping constant is needed to reduce the amplitude by 99% in 1 second? Answer: The damping constant needed to reduce the amplitude by 99% in 1 second is 27.73 kg/s.

Step-by-step solution

Find the decay factor e^(-bt/2m)

Recall that the decay factor for a damped harmonic oscillator is given by \(e^{(-bT / 2m)}\), where \(T\) is the time, \(b\) is the damping constant and \(m\) is the mass. This decay factor represents the percentage of the initial amplitude remaining at time \(T\).

Calculate the time T for amplitude to decay to 1%

To find the time it takes for the amplitude to reach 1% of its initial value, we need to solve the equation \(e^{(-10T / (2*3))} = 0.01\). Divide both sides by the mass and rearrange the equation to get: \(T = -\frac{2m\ln{(0.01)}}{b}\). Then, plug in the given values of \(m\) and \(b\) to get the time. \(T = -\frac{2(3)\ln{(0.01)}}{10} = 9.21\) seconds So, it takes 9.21 seconds for the amplitude to decrease to 1% of its original value.

Calculate the damping constant b for amplitude to decay to 1% in 1 second

Now, we want to find the damping constant \(b\) required to decrease the amplitude by 99% in 1 second. This means the decay factor should be 0.01 at \(T=1\). Using the decay factor equation, we have: \(0.01 = e^{(-b / 6)}\). Rearrange the equation to solve for \(b\). \(b = -6\ln{(0.01)} = 27.73\) kg/s So, the damping constant needed to reduce the amplitude by 99% in 1 second is 27.73 kg/s.

Key concepts

Damping Constant

The damping constant, often represented by the symbol \b\), is a crucial parameter when analyzing the motion of a damped harmonic oscillator. It quantitatively describes the resistance that a medium, such as oil or air, provides against the motion of the object attached to the spring.
When an object oscillates in a resistive medium, the damping constant slows down the object's motion by dissipating its energy, usually in the form of thermal energy. A higher damping constant indicates a stronger resistance, therefore a quicker loss of the oscillator's energy.
For instance, in the exercise, the damping constant of the oil is given as \(10.0\mathrm{~kg/s}\) and is used to determine how long it will take for the amplitude of oscillations to significantly decrease. The value of \(b\) directly affects the time it takes for the oscillation amplitude to decay. In real-world applications, high damping constants are used in systems where rapid stabilization is needed after a disturbance, such as in shock absorbers in vehicles. Conversely, low damping is desired in systems where sustained vibrations are beneficial, such as in the strings of a musical instrument.

Decay Factor

The decay factor is a mathematical expression describing how fast the amplitude of a damped harmonic oscillator decreases over time. Represented as \( e^{(-bT / 2m)} \), where \( T \) is the elapsed time, \( b \) is the damping constant, and \( m \) is the mass of the oscillating object, it reflects the exponential nature of the decay process.
In the steps provided for the solution, the decay factor is used to determine the amount of time needed for the oscillation's amplitude to diminish to a specific percentage of its original value. For example, to calculate the time it takes for the amplitude to drop to 1% of its original value, the decay factor equation is set to 0.01, and the variables are manipulated to solve for time \( T \).
Understanding the decay factor is fundamental for many scientific and engineering fields. It allows the prediction of how different damping constants and masses can influence the rate of energy loss in oscillating systems, which is essential for designing safe and efficient mechanical and structural systems.

Oscillation Amplitude Decay

Oscillation amplitude decay refers to the diminishing magnitude of the oscillations over time in a damped harmonic oscillator. The decay relates to how quickly the peaks of the oscillatory motion decrease, which is governed by the damping constant and the system's characteristics.
In our exercise example, the goal is to determine the time required for the amplitude to decay to just 1% of its initial value. This is indicative of how amplitude decay can be quantified and used to understand the behavior of oscillating systems under the influence of damping.
Real-life examples of oscillation amplitude decay can be seen in pendulums coming to a rest or the steady decline of vibrations in a plucked guitar string. Understanding this concept is essential for tuning systems to optimize performance, such as reducing unwanted vibrations in engines or tuning the response of electronic circuits.