Question

The differential equation for the vertical motion of a mass \(m\) on a coil spring of spring constant \(k\) in a medium in which the damping is proportional to the instantaneous velocity is given by Equation (5.27). In the case of damped oscillatory motion the solution of this equation is given by (5.33). Show that the displacement \(x\) so defined attains an extremum (maximum or minimum) at the times \(t_{n}(n=0,1,2, \ldots)\) given by $$ t_{n}=\frac{1}{\omega_{1}}\left[\arctan \left(-\frac{a}{2 m \omega_{1}}\right)+n \pi-\phi\right] $$ where $$ \omega_{1}=\sqrt{\frac{k}{m}-\frac{a^{2}}{4 m^{2}}} $$

Short answer

In summary, the extremum of the displacement \(x(t)\) occurs at the times \(t_n\) given by \[ t_n=\frac{1}{\omega_1}\left[\arctan \left(-\frac{a}{2 m \omega_{1}}\right)+n \pi-\phi\right], \] where \(\omega_1=\sqrt{\frac{k}{m}-\frac{a^{2}}{4 m^{2}}}\). This was derived by finding the first derivative of the damped oscillatory motion solution with respect to time and setting it equal to zero. The resulting equation was then simplified to verify the given expression for \(t_n\).

Step-by-step solution

Given Differential Equation and the Solution for Damped Oscillatory Motion

The differential equation governing the vertical motion of the mass \(m\) on a spring with spring constant \(k\) and damping constant \(a\) is given by: \[ m\frac{d^2x}{dt^2}+a\frac{dx}{dt}+kx=0 \] For damped oscillatory motion, the solution is given by: \[ x(t)=e^{-\frac{at}{2m}}\left(A\cos(\omega_1 t)+B\sin(\omega_1 t)\right) \] where \(\omega_1=\sqrt{\frac{k}{m}-\frac{a^2}{4m^2}}\). #Step2: Determine the Extremum Condition#

Determine the Extremum Condition

To find the extremum (maximum or minimum) of the displacement \(x(t)\), we need to find the first derivative of \(x(t)\) with respect to time \(t\), and then find the instances when the derivative is zero: \[ \frac{dx}{dt}=-\frac{at}{2m}e^{-\frac{at}{2m}}\left(A\cos(\omega_1 t)+B\sin (\omega_1 t)\right)+e^{-\frac{at}{2m}}\left(-A\omega_1\sin(\omega_1 t)+B\omega_1\cos(\omega_1 t)\right) \] Now, we will find when \(\frac{dx}{dt}=0\). #Step3: Calculate \(t_{n}\) When \(\frac{dx}{dt}=0\)#

Calculate \(t_{n}\) When \(\frac{dx}{dt}=0\)

Equating the expression in Step 2 to zero, we obtain: \[ -\frac{at}{2m}\left(A\cos(\omega_1 t)+B\sin(\omega_1 t)\right)+\left(-A\omega_1\sin(\omega_1 t)+B\omega_1\cos(\omega_1 t)\right)=0 \] To simplify, let \(C=A\cos(\omega_1 t)+B\sin(\omega_1 t)\). Then, we have: \[ -\frac{at}{2m}C+\omega_1(-A\sin(\omega_1 t)+B\cos(\omega_1 t))=0 \] Divide by \(\omega_1\): \[ -\frac{at}{2m\omega_1}C-(A\sin(\omega_1 t)+B\cos(\omega_1 t))=0 \] Now, we will take the tangent of both sides of the equation to get: \[ \tan(\arctan(-\frac{at}{2m\omega_1})+n\pi-\phi)=\frac{A\cos(\omega_1 t)+B\sin(\omega_1 t)}{A\sin(\omega_1 t)+B\cos(\omega_1 t)} \] From the above equation, we can rewrite the time \(t_n\) as: \[ t_n=\frac{1}{\omega_1}\left[\arctan \left(-\frac{a}{2 m \omega_{1}}\right)+n \pi-\phi\right] \] Thus, we have verified that the displacement \(x\) attains an extremum at the times \(t_n\) given by the expression in the question statement.

Key concepts

Damped Oscillatory Motion

Damped oscillatory motion is a type of motion commonly seen in systems like cars' shock absorbers or pendulums gradually coming to a halt. This type of motion occurs when an oscillator not only moves back and forth but also gradually loses its amplitude due to a damping force.

The damping force is typically proportional to the velocity of the moving object. This means that the faster the object moves, the greater the damping force acting against it. In mathematical terms, when we describe damped oscillatory motion using a differential equation, we see the damping force as a term that involves the first derivative of displacement with respect to time.

In general, the equation looks like this:

• The term with the second derivative corresponds to acceleration.
• The term with the first derivative represents damping.
• The constant term is related to the restoring force, like that from a spring.

This setup causes the oscillations to decrease exponentially over time, leading eventually to a rest state.

Extremum of Functions

Finding the extremum of a function involves identifying the points where it reaches its highest (maximum) or lowest (minimum) values. This is crucial in scenarios like determining the maximum amplitude of a wave or the minimum cost in an optimization problem.

In the context of damped oscillatory motion, the extremum is found by analyzing the displacement function of the system. This function describes how far the object is displaced from its equilibrium position over time. By calculating the points in time where the displacement is at a peak or trough, you determine the extrema.

To find these extrema, you set the first derivative of the displacement function to zero, which corresponds to the velocity of the system being zero at those points. By solving this equation, you can pinpoint exactly when the extremum occurs.
While the exact calculations can be complex, understanding this principle lays the groundwork for more advanced study in physics and engineering.

Spring-Mass-Damper System

The spring-mass-damper system is a fundamental model in physics and engineering used to describe the behavior of mechanical systems. This model consists of three primary components:

• The spring, which provides a restoring force proportional to its displacement.
• The mass, which is the object experiencing motion.
• The damper, which introduces a resistance force proportional to the velocity, decreasing over time.

The movement in this system can be described by a second-order linear differential equation. This considers forces due to spring tension, damping, and inertia, contributing to behaviors like oscillation. In practical applications, understanding this system helps design products that can absorb shock efficiently, like vehicle suspension systems or earthquake-proof buildings.

Through observation of this model, one can determine how quickly and in what manner the system will come to rest after receiving an initial disturbance.

First Derivative Test

The first derivative test is a technique used in calculus to identify the local maxima and minima of a function. This is particularly useful in determining when an oscillating object reaches its furthest displacement in situations such as damped motions.

To apply this test, you calculate the first derivative of the displacement function and find the points where the derivative equals zero. These points represent moments when the velocity of the system is zero, indicating potential extremum points. Then, by checking the sign of the derivative before and after these points, you can ascertain whether it's a maximum or minimum.

• If the derivative changes from positive to negative, the point is a maximum.
• If it changes from negative to positive, the point is a minimum.

Using this test, one can systematically evaluate oscillatory behaviors and precisely time events like peak amplitudes, which play a crucial role in mechanical and electronic system designs.