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Chapter 13: Problem 3

What happens to the frequency of a simple harmonic oscillator when the spring constant is doubled? When the mass is doubled?

Short Answer

Expert verified
Doubling the spring constant increases the frequency by a factor of \( \sqrt{2}\), while doubling the mass decreases the frequency by a factor of \( \sqrt{2}\).

Step by step solution

01

Calculate the original frequency

The frequency of the oscillator is given by the formula \(f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}\). Let's suppose that the original values of k and m are denoted by \(k_{1}\) and \(m_{1}\) respectively. So, the original frequency \(f_{1}\) would be: \(f_{1} = \frac{1}{2 \pi} \sqrt{\frac{k_{1}}{m_{1}}}\).
02

Double the spring constant

As per the exercise, the spring constant is doubled. So, the new spring constant will be \(2k_{1}\). Substituting this into the frequency formula, the new frequency \(f_{2}\) would be: \(f_{2} = \frac{1}{2 \pi} \sqrt{\frac{2k_{1}}{m_{1}}}\). By simplifying, we get \(f_{2} = \sqrt{2}f_{1}\). Hence, the frequency will increase by a factor of \( \sqrt{2}\) when the spring constant is doubled.
03

Double the mass

Now, when we double the mass in the same oscillator, the new mass will be \(2m_{1}\). Substituting this into the frequency formula, the new frequency \(f_{3}\) would be: \(f_{3} = \frac{1}{2 \pi} \sqrt{\frac{k_{1}}{2m_{1}}}\). By simplifying, we get \(f_{3} = \frac{f_{1}}{\sqrt{2}}\). Hence, the frequency will decrease by a factor of \( \sqrt{2}\) when the mass is doubled.

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Most popular questions from this chapter

The muscles that drive insect wings minimize the energy needed for flight by "choosing" to move at the natural oscillation frequency of the wings. Biologists study this phenomenon by clipping an insect's wings to reduce their mass. If the wing system is modeled as a simple harmonic oscillator, by what percent will the frequency change if the wing mass is decreased by \(25 \% ?\) Will it increase or decrease?

The vibration frequencies of molecules are much higher than those of macroscopic mechanical systems. Why?

A violin string playing the note A oscillates at \(440 \mathrm{Hz}\). What's its oscillation period?

Explain why the frequency of a damped system is lower than that of the equivalent undamped system.

A 200 -g mass is attached to a spring of constant \(k=5.6 \mathrm{N} / \mathrm{m}\) and set into oscillation with amplitude \(A=25 \mathrm{cm} .\) Determine (a) the frequency in hertz, (b) the period, (c) the maximum velocity, and (d) the maximum force in the spring.
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