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Chapter 13: Problem 15

How can a system have more than one resonant frequency?

Short Answer

Expert verified
A system can have more than one resonant frequency due to the presence of multiple components or parts, each of which could potentially have its own unique natural frequency. When these components are subjected to an external stimulus of a frequency equal to their natural frequencies, they each resonate, leading to multiple resonant frequencies for the entire system.

Step by step solution

01

Understanding Resonance

Resonance is a phenomenon that occurs when an object or system is driven to oscillate at specific frequencies, called resonant frequencies. These are the frequencies at which the system naturally tends to vibrate. When an external stimulus of a frequency equal to the system's natural frequency is applied to the system, the system resonates and its amplitude of oscillation increases dramatically.
02

Complexity of Systems

When we think about complex systems such as an electrical circuit or a mechanical system with many components, there isn't just one natural frequency. Each component, such as an inductor, a capacitor in an electrical circuit, or a spring in a mechanical system, can have its own natural frequency. Therefore, when an external stimulus is applied, each component vibrates at its intrinsic frequency if the conditions allow.
03

Multiple Resonant Frequencies

As a result of different components having different natural frequencies, a system can have more than one resonant frequency. For instance, in an electrical circuit with multiple LC circuits (consisting of an inductor and capacitor), each LC circuit can resonate at a different frequency. The resonant frequencies of a system are the natural frequencies of the individual components that make up the system. Therefore, a system with numerous different components with differing natural frequencies will produce multiple resonant frequencies when the appropriate external stimuli are applied.

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Key Concepts

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

Resonance
Resonance occurs when a system is exposed to an external force that matches one of the system's natural frequencies. This alignment causes the system to oscillate with greater amplitude. Imagine pushing a child on a swing: if you push at the right moment, in sync with the swing's natural rhythm, it swings higher. This is resonance in action. It's fascinating how resonance can amplify vibrations in simple and complex systems alike. When resonance occurs, the energy transfer is at its most efficient, causing objects to vibrate intensely.
Natural Frequency
Every system or object has its own natural frequency, a specific frequency at which it naturally prefers to oscillate. This is like a fingerprint unique to each system. Think about a slinky or a guitar string; each can vibrate at certain sets of frequencies that are unique to their physical properties, such as length, tension, and mass. Natural frequencies are the foundation for understanding resonance. When a system is exposed to a force at this particular frequency, it will resonate, often leading to large oscillations.
Complex Systems
Complex systems are composed of multiple interacting components. These can range from intricate mechanical systems to sophisticated electronic circuits. In such systems, each component might have its own unique natural frequency. Consequently, complex systems can exhibit multiple resonant frequencies. For example, imagine an orchestra where each instrument plays a note that resonates at a different frequency. The complexity arises from the interactions between components, where each part contributes to the overall resonance profile. Such interactions can lead to multiple resonances when stimulated appropriately.
Inductor
An inductor, found in many electrical circuits, is a component that stores energy in a magnetic field when electric current passes through it. Its natural frequency depends on several factors, including its inductance and the rate of change of current. Inductors play a crucial role in LC circuits, where they work with capacitors to determine the resonance of the circuit. When an inductor resonates, it can cause large oscillations in current throughout a circuit. This property makes inductors essential for tuning circuits, such as radios to specific frequencies.
Capacitor
A capacitor is another fundamental component in electrical circuits, responsible for storing energy in an electric field. It acts as a temporary battery, charging up and then discharging at a rate determined by its capacitance. This rate contributes to the natural frequency of an LC circuit when paired with an inductor. Capacitors help control the oscillation and frequency response of electronic circuits. By adjusting the capacitance, you can align the circuit's resonant frequency to a desired level. This functionality is vital for filtering signals and ensuring stable electronic performance.

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Most popular questions from this chapter

Write expressions for simple harmonic motion (a) with amplitude \(10 \mathrm{cm},\) frequency \(5.0 \mathrm{Hz},\) and maximum displacement at \(t=0\) and (b) with amplitude \(2.5 \mathrm{cm},\) angular frequency \(5.0 \mathrm{s}^{-1},\) and maximum velocity at \(t=0\)

The human eye and muscles that hold it can be modeled as a massspring system with typical values \(m=7.5 \mathrm{g}\) and \(k=2.5 \mathrm{kN} / \mathrm{m}\) What's the resonant frequency of this system? Shaking your head at this frequency blurs vision, as the eyeball undergoes resonant oscillations.

Two identical mass-spring systems consist of \(430-\mathrm{g}\) masses on springs of constant \(k=2.2 \mathrm{N} / \mathrm{m} .\) Both are displaced from equilibrium, and the first is released at time \(t=0 .\) How much later should the second be released so their oscillations differ in phase by \(\pi / 2 ?\)

Two mass-spring systems have the same mass and the same total energy. The amplitude of system 1 is twice that of system \(2 .\) How do (a) their frequencies and (b) their maximum accelerations compare?

Explain how simple harmonic motion might be used to determine the masses of objects in an orbiting spacecraft.
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