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Chapter 5: Problem 6

A 4-lb weight is hung upon the lower end of a coil spring hanging vertically from a fixed support. The weight comes to rest in its equilibrium position, thereby stretching the spring 8 in. The weight is then pulled down a certain distance below this equilibrium position and released at \(t=0\). The medium offers a resistance in pounds numerically equal to \(a(d x / d t)\), where \(a>0\) and \(d x / d t\) is the instantaneous velocity in feet per second. Show that the motion is oscillatory if \(a<\sqrt{3}\), critically damped if \(a=\sqrt{3}\), and overdamped if \(a>\sqrt{3}\).

Short Answer

Expert verified
The motion of the weight-spring system depends on the value of the constant \(a\). The motion is oscillatory when \(a < \sqrt{3}\), critically damped when \(a = \sqrt{3}\), and overdamped when \(a > \sqrt{3}\).

Step by step solution

01

Set up the equation of motion

The equation of motion for the system can be set up using Newton's second law, which states that the sum of the forces acting on an object is equal to its mass times its acceleration. We have three forces: the spring force, the resistance force due to the medium, and the gravitational force. Let \(x\) be the displacement from the equilibrium position, and let \(x=0\) correspond to the equilibrium position. The spring force is given by Hooke's law: \(-kx\), where \(k\) is the spring constant. The gravitational force is given by the weight of the object: \(W=4\text{ lb}\). The resistance force due to the medium is given in the problem as: \(-a\frac{dx}{dt}\), where \(\frac{dx}{dt}\) is the velocity of the object. So, the equation of motion is given by: \[m\frac{d^2x}{dt^2} = -kx - a\frac{dx}{dt} - W\]
02

Find the spring constant and mass

We are given the equilibrium conditions, i.e. when the system comes to rest, the spring is stretched 8 in. At this point, the spring force balances the gravitational force: \[kx = W\] Since weight is held constant at 4 lb when the spring is stretched by \(\frac{2}{3}\) ft: \[k(\frac{2}{3}) = 4 \Rightarrow k = 6\] Next, we need to find the mass of the object. Since \(1\text{ lb} = g\text{ slugs}\), where \(g = 32 \frac{\text{ft}}{\text{s}^2}\), we have: \[m = \frac{W}{g} = \frac{4}{32}=\frac{1}{8}\]
03

Rewrite the equation of motion

Now, substitute the values of \(k\) and \(m\) into the equation of motion: \[\frac{1}{8}\frac{d^2x}{dt^2}=-6x-a\frac{dx}{dt}-4\] Rearrange to cancel out the constant term and obtain the following ODE: \[\frac{d^2x}{dt^2} + 8a\frac{dx}{dt} + 48x = 0\]
04

Determine the motion based on the value of a

We have a second-order linear homogeneous ODE, and the behavior of the system depends on the roots of the characteristic equation: \[\lambda^2 + 8a\lambda + 48 = 0\] Now, we use the discriminant \(\Delta\) of the characteristic equation to determine the motion. The discriminant is given by: \[\Delta = (8a)^2 - 4(1)(48) = 64a^2 - 192\] The behavior of the system depends on the value of \(\Delta\): - oscillatory if \(\Delta < 0\) - critically damped if \(\Delta = 0\) - overdamped if \(\Delta > 0\) Now, let's find the conditions on \(a\) based on the discriminant.
05

Find the range of a for different types of motion

a) Oscillatory motion (\(\Delta < 0\)): \[64a^2 - 192 < 0 \Rightarrow a^2 < 3\] b) Critically damped motion (\(\Delta = 0\)): \[64a^2 - 192 = 0 \Rightarrow a^2 = 3\] c) Overdamped motion (\(\Delta > 0\)): \[64a^2 - 192 > 0 \Rightarrow a^2 > 3\] So, the motion is: - oscillatory when \(a < \sqrt{3}\) - critically damped when \(a = \sqrt{3}\) - overdamped when \(a > \sqrt{3}\)

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Key Concepts

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

Oscillatory Motion
Oscillatory motion refers to the movement back and forth around an equilibrium position. In our exercise, the weight on the spring demonstrates oscillatory motion after being displaced and released. If you think about a pendulum or a playground swing, you've seen oscillatory motion in action.

In this scenario, oscillatory motion arises when the system's damping is light, meaning the resistance to motion is small. If the damping coefficient, which we represent with the constant \(a\), is less than a certain critical value (\(a < \sqrt{3}\) in our exercise), the system won't be able to stop the oscillations quickly.

Some characteristics of oscillatory motion include:
  • Repetitive movement around an equilibrium position
  • Energy swapping between kinetic and potential forms
  • Amplitude gradually reducing over time, depending on damping

Remember, light damping means the oscillations continue for a longer time before coming to rest.
Spring Constant
The spring constant, usually denoted by \(k\), essentially measures how stiff a spring is. The higher the spring constant, the more force is needed to stretch or compress the spring by a unit length. It's a measure of the spring's resistance to deformation.

In this problem, we saw that the spring constant was determined using Hooke's Law, which states that the force exerted by a spring is proportional to its extension or compression. Mathematically, it's expressed as \(F = -kx\), where \(F\) is the force, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position.

In the exercise, when we know that a 4-pound weight stretches the spring by 8 inches (or \( \frac{2}{3} \) feet), we use this condition to calculate the spring constant as \(k = 6\) after converting units. This calculation is crucial as it anchors our differential equation by describing the spring's contribution to the system's motion.
Damped Motion
Damped motion describes a situation where the movement of a system is opposed by a force, in this case, through resistance from the medium. This resistance is termed as damping, and it acts against the velocity of the system, gradually reducing its energy and curtailing its oscillations over time. In our problem, damping manifests in the form of a resistance force \(-a \frac{dx}{dt}\).

Depending on the value of \(a\), damping can be categorized into different types:
  • Underdamping: The damping force is weak, allowing oscillations to occur, as long as \(a < \sqrt{3}\).
  • Critical Damping: At \(a = \sqrt{3}\), this damping level ensures the system returns to equilibrium in the shortest possible time without oscillating.
  • Overdamping: When \(a > \sqrt{3}\), damping is strong, causing the system to return to equilibrium slowly without overshooting.

Understanding each damping type is crucial for recognizing how systems react under different resistance scenarios, which is instrumental in various applications like engineering design and control systems.

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Most popular questions from this chapter

A circuit has in series a resistor \(R \Omega\), and inductor of \(L, H\), and a capacitor of C farads. The initial current is zero and the initial charge on the capacitor is \(Q_{0}\) coulombs. (a) Show that the charge and the current are damped oscillatory functions of time if and only if \(R<2 \sqrt{L / C}\), and find the expressions for the charge and the current in this case. (b) If \(R \geq 2 \sqrt{L / C}\), discuss the nature of the charge and the current as functions of time.

A 4 -lb weight is attached to the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 in. At time \(t=0\) the weight is then struck so as to set it into motion with an initial velocity of \(2 \mathrm{ft} / \mathrm{sec}\), directed downward. (a) Determine the resulting displacement and velocity of the weight as functions of the time. (b) Find the amplitude, period, and frequency of the motion. (c) Determine the times at which the weight is \(1.5\) in. below its equilibrium position and moving downward. (d) Determine the times at which it is \(1.5\) in. below its equilibrium position and moving upward.

A circuit has in series a constant electromotive force of \(40 \mathrm{~V}\), a resistor of \(10 \Omega\), and an inductor of \(0.2 \mathrm{H}\). If the initial current is 0 , find the current at time \(t>0\).

A 20-lb weight is attached to the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 in. Various external forces of the form \(F(t)=\) cos wt are applied to the system and it is found that the resonance frequency is \(0.5\) cycles/sec. Assuming that the resistance of the medium in pounds is numerically equal to \(a(d x / d t)\), where \(d x / d t\) is the instantaneous velocity in feet per second, determine the damping coefficient \(a\).

A circuit has in series an electromotive force given by \(E(t)=5 \sin 100 t \mathrm{~V}\), a resistor of \(10 \Omega\), an inductor of \(0.05 \mathrm{H}\), and a capacitor of \(2 \times 10^{-4}\) farads. If the initial current and the initial charge on the capacitor are both zero, find the charge on the capacitor at any time \(t>0\).
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