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Chapter 16: Problem 58

58\. Lightly Damped The amplitude of a lightly damped oscillator decreases by \(3.0 \%\) during each cycle. What fraction of the mechanical energy of the oscillator is lost in each full oscillation?

Short Answer

Expert verified
The fraction of mechanical energy lost per cycle is 0.0591 or 5.91%.

Step by step solution

01

Understand the problem

Determine the fraction of mechanical energy lost in each full oscillation of a lightly damped oscillator given that the amplitude decreases by 3.0% per cycle.
02

Relate amplitude to energy

The energy of the oscillator is proportional to the square of its amplitude. If the amplitude decreases to 97% of its original value (A_{new} = 0.97A_{original}), then the energy decreases to (0.97^2 = 0.9409) of its original value.
03

Calculate energy loss fraction

The fraction of energy lost in one cycle is the difference between the initial energy and the energy after one cycle: 1 - 0.9409 = 0.0591.
04

Express the result

The fraction of mechanical energy lost in each full oscillation is 0.0591, or in percentage form, 5.91%.

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Key Concepts

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

mechanical energy
Mechanical energy in an oscillator refers to the sum of its potential and kinetic energy. For a simple harmonic oscillator, this energy is stored in the form of a spring's potential energy and the moving mass's kinetic energy.
When the system is undamped, the mechanical energy remains constant with time. This is because energy transitions back and forth between kinetic and potential forms without any outside force affecting it.
In a damped oscillator, however, energy is gradually lost over time because of non-conservative forces, like friction or air resistance, which convert some of the mechanical energy into heat or other non-mechanical forms.
damping in oscillators
Damping in oscillators is a phenomenon where the amplitude of oscillation decreases over time due to external forces like friction. This is important to understand because damping affects the longevity and stability of the oscillatory motion.
There are generally three types of damping:
  • Light Damping: Amplitude decreases gradually over time.
  • Critical Damping: System returns to equilibrium without oscillating.
  • Over Damping: System returns to equilibrium slowly without oscillations.

In the given exercise, the oscillator is lightly damped, meaning its amplitude decreases by a small, consistent percentage each cycle.
amplitude and energy relationship
The energy of an oscillator is proportional to the square of its amplitude. Mathematically, if the amplitude (A) is reduced, the mechanical energy (E) is reduced by the factor of the amplitude squared.
For instance, if the amplitude decreases to 97% of its original value, the new energy will be \(0.97^2\) times the initial energy.
In our exercise, a decrease in amplitude by 3% per cycle means the new energy is 0.9409 times the original energy.
oscillator cycles
An oscillator cycle refers to a full sequence of motion from the starting position, through the extreme positions, and back to the starting position. One cycle is critical for understanding changes in mechanical energy and damping effects.
Each cycle in a lightly damped oscillator exhibits a slight decrease in amplitude and mechanical energy.
In the exercise, each cycle sees a 3% reduction in amplitude, translating to a 5.91% loss in mechanical energy.
energy dissipation
Energy dissipation in a damped oscillator is a crucial concept that explains how energy is lost each cycle. This loss is converted into other forms, primarily thermal energy due to internal friction or air resistance.
The fraction of energy lost can be computed by comparing the initial and reduced energy levels after each cycle.
In our example, the energy at the end of each cycle is 94.09% of the initial energy, leading to a 5.91% loss in mechanical energy with each oscillation full cycle.

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Most popular questions from this chapter

38\. Solid Disk In Fig. \(16-38\), a physical pendulum consists of a uniform solid disk (of mass \(M\) and radius \(R\) ) supported in a vertical plane by a pivot located a distance \(d\) from the center of the disk. The disk is displaced by a small angle and released. Find an expression for the period of the resulting simple harmonic motion.

54\. Particle Undergoing SHM A \(10 \mathrm{~g}\) particle is undergoing simple harmonic motion with an amplitude of \(2.0 \times 10^{-3} \mathrm{~m}\) and a maximum acceleration of magnitude \(8.0 \times 10^{-3} \mathrm{~m} / \mathrm{s}^{2}\). The phase constant is \(-\pi / 3\) rad. (a) Write an equation for the force on the particle as a function of time. (b) What is the period of the motion? (c) What is the maximum speed of the particle? (d) What is the total mechanical energy of this simple harmonic oscillator?

19\. Oscillator An oscillator consists of a block attached to a spring \((k=400 \mathrm{~N} / \mathrm{m}) .\) At some time \(t\), the position (measured from the system's equilibrium location), velocity, and acceleration of the block are \(x=0.100 \mathrm{~m}, v=-13.6 \mathrm{~m} / \mathrm{s}\), and \(a=-123 \mathrm{~m} / \mathrm{s}^{2} .\) Calcu- late (a) the frequency of oscillation, (b) the mass of the block, and (c) the amplitude of the motion.

48\. Large Slingshot A (hypothetical) large slingshot is stretched \(1.50 \mathrm{~m}\) to launch a \(130 \mathrm{~g}\) projectile with speed sufficient to escape from Earth \((11.2 \mathrm{~km} / \mathrm{s})\). Assume the elastic bands of the slingshot obey Hooke's law. (a) What is the spring constant of the device, if all the elastic potential energy is converted to kinetic energy? (b) Assume that an average person can exert a force of \(220 \mathrm{~N}\). How many people are required to stretch the elastic bands?

46\. Angular Amplitude For a simple pendulum, find the angular amplitude \(\Theta\) at which the restoring torque required for simple harmonic motion deviates from the actual restoring torque by \(1.0 \%\). (See "Trigonometric Expansions" in Appendix E.)
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