Question

A damped harmonic oscillator involves a block (m = \(1.91 \mathrm{~kg}\) ), a spring \((k=12.6 \mathrm{~N} / \mathrm{m})\), and a damping force \(F=\) \(-b v_{x} .\) Initially, it oscillates with an amplitude of \(26.2 \mathrm{~cm}\); because of the damping, the amplitude falls to three-fourths of this initial value after four complete cycles. (a) What is the value of \(b ?(b)\) How much energy has been "lost" during these four cycles?

Short answer

The damping factor \(b\) is approximately \(0.474 Ns/m\) and the energy lost during the four cycles of oscillation is \(0.189 J\)

Step-by-step solution

Calculate natural frequency

Firstly, one needs to get the natural frequency \(\omega_{0}\) of the system using the equation \(\omega_{0} = sqrt{k / m}\). Here, \(k\) is the spring constant, \(12.6 N/m\), and \(m\) is the mass, \(1.91 kg\).

Get the angular logarithmic decrement

Next, we can compute the logarithmic decrement which represents the rate of decay of motion in underdamped systems. It can be calculated by \(Δ = ln(A_n / A_{n+1})\), where \(A_n\) and \(A_{n+1}\) are successive amplitudes. Here, \(A_n\) is the initial amplitude which is \(0.262m\), and the amplitude after one cycle \(A_{n+1}\) is \(0.75 * A_n = 0.75 * 0.262 = 0.1965m\). After four cycles, \(4 Δ = ln(A_0 / A_{4}) = ln(0.262 / 0.1965) = 0.2924\). Solve for \(Δ\) to find \(Δ = 0.2924 / 4 = 0.0731\).

Calculate the damping factor \(b\)

The damping constant \(b\) can be found using the formula \(\b = 2 m ζ ω_{0}\). However, solving for the damping coefficient \(b\) requires us to find \(ζ\). \(ζ\) can be obtained from the equation \(ζ = 1 / (2Q)\), where \(Q\) is the quality factor of the oscillator which can be computed by \(Q = π / Δ = π / 0.0731 = 43.082\). Therefore, \(ζ = 1 / (2(43.082)) = 0.0116\). The damping factor \(b\) can then be calculated as \(b = 2 m ζ ω_{0} = 2 * 1.91 kg * 0.0116 * sqrt(12.6 N/m / 1.91 kg) = 0.474 Ns/m\).

Calculate lost energy

The initial energy of system is \(1/2 * k * A_{0}^2 = 1/2 * 12.6 N/m * (0.262 m)^2 = 0.432 J\). After four cycles the energy is \(1/2 * k * A_{4}^2 = 1/2 * 12.6 N/m * (0.1965 m)^2 = 0.243 J\). Therefore, the energy loss in the four cycles is the difference between the initial and final energies, \(0.432 J - 0.243 J = 0.189 J\)

Key concepts

Damping Force

A damping force in a harmonic oscillator system is a force that opposes the motion of the swinging body and causes the amplitude of the oscillations to gradually decrease over time. It is generally modeled by the equation \( F = -b v_{x} \), where \( b \) is the damping coefficient and \( v_{x} \) is the velocity of the oscillating body.

This force is crucial in many real-world systems to prevent continuous and uncontrolled oscillations.#### Importance of Damping Force- **Energy Dissipation**: The damping force removes energy from the system, primarily converting this energy into heat, thereby reducing the mechanical energy stored in the spring.- **System Stability**: By slowing down and eventually stopping oscillations, damping forces help to stabilize systems and prevent mechanical failure or damage.- **Controlling Vibration**: In vehicles and buildings, adequate damping ensures that vibrations do not cause discomfort or structural issues.Understanding the mechanics of the damping force improves comprehension of how oscillations behave in practical engineering and physics problems.

Spring Constant

The spring constant, denoted by \( k \), is a measure of the stiffness of a spring. It is an intrinsic property of the spring itself, indicating how much force is needed to stretch or compress it by a certain amount. Mathematically, it is defined by Hooke's Law, \( F = -k x \), where \( F \) is the force applied to the spring, and \( x \) is the displacement from the equilibrium position.

In the context of a damped harmonic oscillator, the spring constant is pivotal in determining both the natural frequency and the energy stored within the system.#### Key Characteristics of Spring Constant- **Relation with Natural Frequency**: The natural frequency \( \omega_{0} \) of an oscillator is directly dependent on the spring constant, calculated as \( \omega_{0} = \sqrt{\frac{k}{m}} \), where \( m \) is the mass attached to the spring.- **Units**: It is usually represented in Newtons per meter (N/m), indicating the force required for a unit displacement.- **Impact on System Behavior**: A stiffer spring (higher \( k \)) results in higher natural frequency but requires more energy to compress or extend.A detailed understanding of the spring constant is essential in designing systems that rely on precise control of mechanical energies, like shocks in vehicles or timekeeping in watches.

Logarithmic Decrement

Logarithmic decrement is a dimensionless parameter that measures the rate of decay of oscillations in a damped harmonic oscillator. Specifically, it reflects how quickly the amplitude of oscillations decreases over successive cycles. Calculated with the formula: \( \Delta = \ln(\frac{A_n}{A_{n+1}}) \)where \( A_n \) and \( A_{n+1} \) are successive amplitudes. Essentially, it gives us a quantitative insight into damping efficiency.

#### Importance of Logarithmic Decrement- **Understanding Damping Quality**: A larger logarithmic decrement indicates more significant damping, which is useful for assessing how quickly oscillations are suppressed.- **Practical Applications**: Engineers use it to design building structures and automotive suspensions, ensuring they have the optimal balance between stability and comfort.- **Complexity of Systems**: Enables estimation of the quality factor \( Q \), which further elucidates the bandwidth of the oscillatory system: \( Q = \frac{\pi}{\Delta} \).By evaluating logarithmic decrement, scientists and engineers can fine-tune systems to achieve desired performance, balancing between over-damped (slow response) and under-damped (excessive oscillations) behaviors.

Natural Frequency

Natural frequency, denoted as \( \omega_{0} \), is the frequency at which a system oscillates when not subjected by any external force other than the restoring force. It is calculated using the formula \( \omega_{0} = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass. In a damped harmonic oscillator, the natural frequency is a vital parameter that helps in understanding how the system behaves when initially set into motion.

#### Key Insights into Natural Frequency- **Role in Oscillator Design**: Natural frequency helps determine the appropriateness of a spring or system for particular applications, like oscillations in clocks.- **Relation with Damping**: Though the presence of damping adjusts the observed period and frequency of oscillation, the natural frequency serves as a baseline reference.- **Engineering Applications**: In automotive and aerospace industries, knowing the natural frequency ensures that machinery operates smoothly without unwanted resonances that can lead to failure.Natural frequency forms the foundation for analyzing dynamic systems in mechanical and acoustical engineering, simplifying complex behavior into predictable patterns.