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Chapter 13: Problem 5

How does the frequency of a simple harmonic oscillator depend on its amplitude?

Short Answer

Expert verified
The frequency of a simple harmonic oscillator does not depend on its amplitude.

Step by step solution

01

Define the properties

The amplitude \(A\) of a simple harmonic oscillator is the maximum extent of a vibration or oscillation, in other words, the maximum displacement from the equilibrium position. Frequency \(f\), on the other hand, refers to how often a complete oscillation or vibration occurs in a given unit of time. Essentially, it's the inverse of the period \(T\). In a simple harmonic oscillator, the period \(T\) (and hence, the frequency \(f\)) is given by \(T = 2\pi\sqrt{\frac{m}{k}}\) and \(f = \frac{1}{T}\) where \(m\) is the mass and \(k\) is the spring constant.
02

Analyze the relationship

From the equations of \(T\) and \(f\), we can see that neither the period \(T\) nor the frequency \(f\) depends on amplitude \(A\). This indicates that regardless of how large or small the amplitude is, it has no impact on the frequency of a simple harmonic oscillator.
03

Understanding physically

Physically, this makes sense if we understand that the frequency of an oscillation is determined by the properties of the oscillator itself (i.e., its mass and spring constant in this case), not by how much it is initially displaced. Even if the oscillator is displaced more from its equilibrium position, (i.e amplitude increases), it doesn't make the oscillator move faster or slower. The frequency is a fixed property based on its physical characteristics.

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Most popular questions from this chapter

A particle of mass \(m\) has potential energy given by \(U=a x^{2}\) where \(a\) is a constant and \(x\) is the particle's position. Find an expression for the frequency of simple harmonic oscillations this particle undergoes.

A pendulum of length \(L\) is mounted in a rocket. Find its period if the rocket is (a) at rest on its launch pad; (b) accelerating upward with acceleration \(a=\frac{1}{2} g ;\) (c) accelerating downward with \(a=\frac{1}{2} g ;\) and \((d)\) in free fall.

The equation for an ellipse is \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1 .\) Show that two-dimensional simple harmonic motion whose components have different amplitudes and are \(\pi / 2\) out of phase gives rise to elliptical motion. How are constants \(a\) and \(b\) related to the amplitudes?

The protein dynein powers the flagella that propel some unicellular organisms. Biophysicists have found that dynein is intrinsically oscillatory, and that it exerts peak forces of about \(1.0 \mathrm{pN}\) when it attaches to structures called microtubules. The resulting oscillations have amplitude \(15 \mathrm{nm}\). (a) If this system is modeled as a mass-spring system, what's the associated spring constant? (b) If the oscillation frequency is \(70 \mathrm{Hz}\), what's the effective mass?

A doctor counts 77 heartbeats in 1 minute. What are the corresponding period and frequency?
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