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Chapter 12: Problem 4

A harmonic oscillator has a mass \(m=0.300 \mathrm{~kg}\) and a force constant \(k=155 \mathrm{~N} \mathrm{~m}^{-1}\). (a) Find the period and the frequency of oscillation. (b) Find the value of the friction constant \(\zeta\) necessary to produce critical damping with this oscillator. Find the value of the constant \(\lambda_{1}\). (c) Construct a graph of the position of the oscillator as a function of \(t\) for the initial conditions \(z(0)=0, v_{z}(0)=0.100 \mathrm{~m} \mathrm{~s}^{-1} .\)

Short Answer

Expert verified
The period of oscillation is \(T = 2\pi\sqrt{\frac{0.300}{155}}\) s and the frequency is \(f = \frac{1}{T}\). The friction constant that produces critical damping is \(\zeta = 2\sqrt{155 \times 0.300}\) kg/s and the value of \(\lambda_1\) is \(\lambda_1 = \frac{\zeta}{2 \times 0.300}\) s鈦宦�. The position of the oscillator as a function of time is \(z(t) = 0\), thus the plot of \(z\) over \(t\) is a straight line along the time axis.

Step by step solution

01

Finding the Period and Frequency of Oscillation

The period of oscillation is given by \(T = 2\pi\sqrt{\frac{m}{k}}\). Substituting \(m = 0.300\) kg and \(k = 155\) N/m into the equation yields \(T = 2\pi\sqrt{\frac{0.300}{155}}\) s. The frequency \(f\) is then found by taking the inverse of the period. Thus, \(f = \frac{1}{T}\).
02

Finding the Value of the Friction Constant and the Constant \(\lambda_1\)

The value of the friction constant \(\zeta\) that produces critical damping is given by \(\zeta = 2\sqrt{km}\). Substituting \(k = 155\) N/m and \(m = 0.300\) kg, we get \(\zeta = 2\sqrt{155 \times 0.300}\) kg/s. The value of the constant \(\lambda_1\) is then found by using the equation \(\lambda_1 = \frac{\zeta}{2m}\). Substituting \(\zeta\) and \(m\), we get \(\lambda_1 = \frac{\zeta}{2 \times 0.300}\) s鈦宦�.
03

Constructing the Graph of the Position of the Oscillator as a Function of time

For an oscillator with initial conditions \(z(0) = 0\) and \(v_{z}(0) = 0.100\) m/s, the position of the oscillator as a function of time under critical damping is represented by the equation \(z(t) = Ce^{-\lambda_1t}\). Given that \(z(0) = 0\), it can be deduced that \(C = 0\) and therefore \(z(t) = 0\) for all \(t\). Therefore, a graph plotting \(z\) over \(t\) would be a straight line along the time axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period of Oscillation
The period of oscillation, often denoted by \( T \), is a fundamental concept in the study of harmonic oscillators. It represents the time taken for the oscillator to complete one full cycle of motion. In mathematical terms, for a simple harmonic oscillator, the period can be calculated using the formula:\[ T = 2\pi \sqrt{\frac{m}{k}} \]where \( m \) is the mass of the oscillator and \( k \) is the force constant of the spring. This formula highlights a key characteristic of harmonic motion: the period is independent of the amplitude of oscillation, solely depending on the mass and the spring constant.
  • A larger mass will result in a longer period, as the system takes more time to complete an oscillation.
  • A stiffer spring (higher \( k \)) reduces the period, leading to faster oscillations.
The simplicity of this relationship makes it a crucial tool in understanding various physical systems, from simple springs to more complex systems like pendulums.
Frequency of Oscillation
Frequency of oscillation, denoted by \( f \), is another important concept that helps describe the behavior of a harmonic oscillator. It is defined as the number of complete oscillations performed per unit of time, usually seconds.The relationship between the frequency and the period is inversely proportional, given by the formula:\[ f = \frac{1}{T} \]where \( T \) is the period of oscillation. This implies that as the period increases, the frequency decreases, and vice versa.
  • If an oscillator has a short period, it produces more cycles in a given amount of time, resulting in a higher frequency.
  • Conversely, a long period signifies fewer cycles per unit time, thus a lower frequency.
An everyday example is the motion of a clock pendulum where the consistent ticking is a manifestation of regular oscillation frequency. Understanding these dynamics is essential in fields like acoustics, mechanics, and electronics, where oscillations play a pivotal role.
Critical Damping
Critical damping is a concept that emerges when discussing the damping of oscillators, especially in systems that are designed to stop oscillating as quickly as possible without oscillating.Damping occurs due to resistive forces like friction, which can be characterized by the friction constant \( \zeta \). Critical damping is achieved when the damping force is optimized to bring the system back to equilibrium in the shortest possible time:\[ \zeta = 2\sqrt{km} \]where \( k \) is the spring constant and \( m \) is the mass of the oscillator. With critical damping:
  • The system returns to equilibrium as fast as possible without overshooting.
  • It's the ideal level of damping for many applications, like automotive shock absorbers, which provide a smooth ride by preventing oscillations.
In practical terms, finding the correct value of \( \zeta \) is vital in engineering to avoid oscillations that could lead to structural damage over time.

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Most popular questions from this chapter

An object of mass \(m\) is subjected to a gradually increasing force given by \(F_{0}\left(1-e^{-b t}\right)\) where \(a\) and \(b\) are constants. Solve the equation of motion of the particle. Find the particular solution for the case that \(x(0)=0\) and \(\mathrm{d} x / \mathrm{d} t=0\) at \(t=0\).

A tank contains a solution that is rapidly stirred, so that it remains uniform at all times. A solution of the same solute is flowing into the tank at a fixed rate of flow, and an overflow pipe allows solution from the tank to flow out at the same rate. If the solution flowing in has a fixed concentration that is different from the initial concentration in the tank, write and solve the differential equation that governs the number of moles of solute in the tank. The inlet pipe allows \(A\) moles per hour to flow in and the overflow pipe allows \(B n\) moles per hour to flow out, where \(A\) and \(B\) are constants and \(n\) is the number of moles of solute in the tank. Find the values of \(A\) and \(B\) that correspond to a volume in the tank of \(100.01\), an input of \(1.0001 \mathrm{~h}^{-1}\) of a solution with \(1.000 \mathrm{~mol} 1^{-1}\), and an output of \(1.0001 \mathrm{~h}^{-1}\) of the solution in the tank. Find the concentration in the tank after \(5.00 \mathrm{~h}\), if the initial concentration is zero.

A less than critically damped harmonic oscillator has a mass \(m=0.3000 \mathrm{~kg}\), a force constant \(k=98.00 \mathrm{~N} \mathrm{~m}^{-1}\), and a friction constant \(\zeta=1.000 \mathrm{~kg} \mathrm{~s}^{-1}\). (a) Find the circular frequency of oscillation \(\omega\) and compare it with the frequency that would occur if there were no damping. (b) Find the time required for the real exponential factor in the solution to drop to one-half of its value at \(t=0\).

An \(n\) th-order chemical reaction with one reactant obeys the differential equation $$ \frac{\mathrm{d} c}{\mathrm{~d} t}=-k c^{n}, $$ where \(c\) is the concentration of the reactant and \(k\) is a constant. Solve this differential equation by separation of variables. If the initial concentration is \(c_{0}\) moles per liter, find an expression for the time required for half of the reactant to react.

A forced harmonic oscillator with mass \(m=0.300 \mathrm{~kg}\) and a circular frequency \(\omega=6.283 \mathrm{~s}^{-1}\) (frequency \(v=\) \(1.000 \mathrm{~s}^{-1}\) ) is exposed to an external force \(F_{0} \exp (-\) \(\beta t) \sin (\alpha t)\) with \(\alpha=7.540 \mathrm{~s}^{-1}\) and \(\beta=0.500 \mathrm{~s}^{-1}\). Find the solution to its equation of motion. Construct a graph of the motion for several values of \(F_{0}\).
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