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Chapter 11: Problem 32

(a) If a vibrating system has total energy \(E_{0},\) what will its total energy be (in terms of \(E_{0}\) ) if you double the amplitude of vibration? (b) If you want to triple the total energy of a vibrating system with amplitude \(A_{0}\), what should its new amplitude be (in terms of \(A_{0}\) )?

Short Answer

Expert verified
(a) The total energy becomes 4\(E_{0}\) if amplitude is doubled. (b) New amplitude should be \(\sqrt{3}A_{0}\) to triple the energy.

Step by step solution

01

Understanding the Relationship

For a vibrating system, the total energy is proportional to the square of its amplitude. If \(E\) is the total energy and \(A\) is the amplitude, then \(E \propto A^2\). Therefore, we can express this as \(E = kA^2\), where \(k\) is a constant.
02

Solving Part (a) 鈥� Doubling the Amplitude

If the amplitude is doubled from \(A_{0}\) to \(2A_{0}\), the new energy \(E'\) becomes \(E' = k(2A_{0})^2 = k \cdot 4A_{0}^2\). Since the original energy \(E_{0} = kA_{0}^2\), replacing \(kA_{0}^2\) in the expression gives \(E' = 4E_{0}\). Thus, the new total energy is four times the original energy.
03

Solving Part (b) 鈥� Tripling the Energy

To triple the energy, the new energy \(E' = 3E_{0}\). Using the equation \(E = kA^2\), we need \(kA^2 = 3kA_{0}^2\). Simplifying gives \(A^2 = 3A_{0}^2\), leading to \(A = \sqrt{3}A_{0}\). Thus, to triple the energy, the amplitude should be \(\sqrt{3}\) times the original amplitude.

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Key Concepts

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

Amplitude of Vibration
Amplitude of vibration refers to the maximum distance a point on a wave, such as a sound wave or an oscillating object, moves from its rest position. Think of it like the height of a wave peak. In a vibrating system, amplitude determines how strong or loud the vibration is.
  • A high amplitude means a stronger or more intense vibration.
  • A low amplitude indicates weaker vibrations.
Note that amplitude is different from frequency, which measures how often vibrations occur. Understanding amplitude is crucial in solving problems involving vibrating systems. It sets the stage for understanding the relationship between amplitude and energy.
Energy Proportionality
In physics, energy proportionality means that one physical quantity changes proportionally with another. For vibrating systems, energy is proportional to the square of the amplitude.
This relationship can be expressed with the formula:
\[ E = kA^2 \]
Here,
  • \(E\) represents the total energy.
  • \(A\) is the amplitude.
  • \(k\) is a constant related to the system's properties.

This equation indicates that even a small change in amplitude causes a significant energy change. If amplitude doubles, energy quadruples. This proportionality is a key concept for grasping how mechanical systems behave.
Physics Problem Solving
Physics problem solving often involves applying formulas and understanding core principles. When tackling problems about vibrating systems, start by identifying known variables and relationships, like we did with energy and amplitude.
  • Begin by defining the problem clearly. What are you trying to find out?
  • Identify known and unknown variables.
  • Apply relevant formulas, such as \(E = kA^2\).
  • Use logical reasoning to manipulate equations and solve for unknowns.
Breaking down a problem step-by-step helps in reaching a solution efficiently. Remember, practice is key to becoming proficient at problem solving in physics.
Vibrating System
A vibrating system is any system capable of oscillations or repetitive motions. These systems can include musical instruments, mechanical springs, and even structures like bridges.
Key components of a vibrating system often include:
  • Mass: This provides the inertia needed for motion.
  • Spring or Elastic Material: Usually presents a restoring force.
  • Energy: Required to maintain vibrations.
Vibration in such systems can be free, where it continues without sustained external influence, or forced, where an external force maintains the movement. Principles like energy proportionality are essential for understanding how these systems behave under different conditions.
Energy and Amplitude Relationship
The energy and amplitude relationship in a vibrating system reveals how changes in amplitude affect energy levels. This relationship is crucial in designing efficient systems, such as in engineering and acoustics.
  • If the amplitude of a system is doubled, its energy becomes fourfold due to the \(E = kA^2\) relationship.
  • Conversely, if energy needs to be tripled, the amplitude must be increased by \(\sqrt{3}\) times the original.
Understanding this relationship helps in predicting the effects of modifying a system's amplitude, aiding in both problem-solving and practical applications. This makes amplitude and energy considerations key in fields like mechanical engineering and sound design.

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Most popular questions from this chapter

A concrete block is hung from an ideal spring that has a force constant of \(200 \mathrm{~N} / \mathrm{m} .\) The spring stretches \(0.120 \mathrm{~m} .\) (a) What is the mass of the block? (b) What is the period of oscillation of the block if it is pulled down \(1.0 \mathrm{~cm}\) and released? (c) What would be the period of oscillation if the block and spring were placed on the moon?

A steel wire has the following properties: $$ \begin{array}{l} \text { Length }=5.00 \mathrm{~m} \\ \text { Cross-sectional area }=0.040 \mathrm{~cm}^{2} \end{array} $$ Young's modulus \(=2.0 \times 10^{11} \mathrm{~Pa}\) Shear modulus \(=0.84 \times 10^{11} \mathrm{~Pa}\) Proportional limit \(=3.60 \times 10^{8} \mathrm{~Pa}\) Breaking stress \(=11.0 \times 10^{8} \mathrm{~Pa}\) The proportional limit is the maximum stress for which the wire still obeys Hooke's law (see point \(\mathrm{B}\) in Figure 11.12 ). The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit? (b) How much does the wire stretch under this load? (c) What is the maximum weight that can be supported?

A toy is undergoing SHM on the end of a horizontal spring with force constant \(300.0 \mathrm{~N} / \mathrm{m} .\) When the toy is \(0.120 \mathrm{~m}\) from its equilibrium position, it is observed to have a speed of \(3 \mathrm{~m} / \mathrm{s}\) and a total energy of \(4.4 \mathrm{~J}\). Find (a) the mass of the toy, (b) the amplitude of the motion, and (c) the maximum speed attained by the toy during its motion.

A harmonic oscillator is made by using a \(0.600 \mathrm{~kg}\) frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of \(0.150 \mathrm{~s}\) and a maximum speed of \(2 \mathrm{~m} / \mathrm{s} .\) Find \((\mathrm{a})\) the force constant of the spring and \((\mathrm{b})\) the amplitude of the oscillation.

(a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern having the same frequency of the note that is sung. If someone sings the note \(\mathrm{B}\) flat that has a frequency of 466 \(\mathrm{Hz}\), how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of \(50.0 \mu \mathrm{s}\). What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from \(2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) to \(4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as X-rays do. To detect small objects such as tumors, a frequency of around \(5.0 \mathrm{MHz}\) is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?
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