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Chapter 22: Problem 2

Write down the differential equation describing undamped free vibrations, its general solution and the definitions of 'amplitude,' 'period,' and 'frequency.'

Short Answer

Expert verified
The differential equation for undamped free vibrations is \( m\frac{d^2x}{dt^2} + kx = 0 \). The general solution is \( x(t) = A\cos(\omega t + \phi) \). Amplitude is the maximum displacement from equilibrium, period is the time for one vibration cycle, and frequency is the number of cycles per second.

Step by step solution

01

Write down the differential equation for undamped free vibrations

Undamped free vibrations can be modeled by a second-order ordinary differential equation without a damping term. The most basic form of this equation is given by Newton's second law for a mass-spring system, which is: \( m\frac{d^2x}{dt^2} + kx = 0 \), where \(m\) is the mass of the object, \(x\) is the displacement from equilibrium, \(k\) is the spring constant, and \(t\) is time.
02

Present the general solution of the differential equation

The general solution to the differential equation \( m\frac{d^2x}{dt^2} + kx = 0 \) is an oscillating function that can be represented as: \( x(t) = A\cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency \(\small{(\omega = \sqrt{\frac{k}{m})}}\), \( t \) is time, and \( \phi \) is the phase angle.
03

Define Amplitude

Amplitude (\(A\)) is defined as the maximum displacement of the object from its equilibrium position. It represents the maximum extent of the vibration and is a positive quantity, which can be measured in meters (m) in the case of mechanical vibrations.
04

Define Period

The period (\(T\)) of the oscillation is the time it takes for one complete cycle of the vibration. Mathematically, it is related to the angular frequency by \( T = \frac{2\pi}{\omega} \).
05

Define Frequency

Frequency (\(f\)) is defined as the number of complete cycles of vibration occurring in one second. It is the reciprocal of the period, given by \( f = \frac{1}{T} = \frac{\omega}{2\pi} \). The unit of frequency is Hertz (Hz).

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Key Concepts

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

Differential Equation for Undamped Vibrations
Understanding the motion of an object in undamped free vibration is key to numerous applications in physics and engineering. This concept is described mathematically using a second-order ordinary differential equation:

\[ m\frac{d^2x}{dt^2} + kx = 0 \],

where 'm' represents the mass of the vibrating object, 'k' is the spring constant, and 'x' is the displacement from the object's equilibrium position. The absence of a damping term in this equation is what characterizes undamped vibrations. Such a system, if displaced from its equilibrium position and then released, will continue to oscillate indefinitely with a constant amplitude.
General Solution of Oscillation
The general solution to the undamped vibration equation provides insight into how the system behaves over time. It is expressed as:

\[ x(t) = A\cos(\omega t + \phi) \],

where 'A' denotes the amplitude, '\(\omega\)' is the angular frequency, 't' is time, and '\(\phi\)' is the initial phase angle. This trigonometric function captures the oscillatory nature of the system, showing that the displacement 'x' varies as a cosine function of time, allowing us to predict the position of the object at any given moment.
Amplitude Definition
The term 'amplitude' might seem abstract, but it simply refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. Imagine swinging on a swing - the highest point you reach is comparable to the amplitude of an oscillating system. In our equation:

\[ A \],

it signifies how far from rest the object moves during its oscillation and is always a positive value. Whether you're considering sound waves, ocean waves, or a mass on a spring, understanding amplitude is crucial as it determines the energy stored in the oscillation.
Period of Oscillation
The 'period of oscillation' is an important concept, especially when dealing with repetitive motion. Defined as the time needed to complete one full cycle of the vibration, the period, denoted by \(T\), provides a link to the motion's repetitiveness. In terms of the angular frequency '\(\omega\)', the relationship is:

\[ T = \frac{2\pi}{\omega} \].

This measures the time interval between successive instances when the object passes through a specific point in the same direction.
Vibrational Frequency
Frequency, often symbolized as 'f', turns our conversation to how often something happens. For vibrations, it's the count of how many cycles occur per second. Specifically:

\[ f = \frac{1}{T} = \frac{\omega}{2\pi} \],

it is the rate at which the object completes its vibratory cycles. Expressed in Hertz (Hz), frequency is inversely related to the period, meaning that the smaller the period (faster oscillation), the higher the frequency, and vice versa. This is a fundamental concept in wave mechanics, as frequency affects everything from the pitch of a sound to the color of light.

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Most popular questions from this chapter

In damped free vibrations of a system, when is it the case that vibrations die out in time?

Why is it possible to use conservation of energy to obtain the natural frequency of a body executing undamped free vibrations.?
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