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Chapter 13: Problem 72

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?

Short Answer

Expert verified
(a) Frequencies of both systems are the same. (b) The maximum acceleration of System 1 is twice that of System 2.

Step by step solution

01

Identify the Known Variables and Concepts

The amplitude for System 1 is twice that of System 2. Both systems have the same mass and total energy. Total energy in a mass-spring system is kinetic + potential, and the potential energy stored in a spring is \( \frac{1}{2} kA^2 \), where \( k \) is the spring constant and \( A \) is the amplitude. Additionally, the frequency of the system is given by \( \sqrt{\frac{k}{m}} \), where \( m \) is the mass. Finally, the maximum acceleration in a mass-spring system occurs at the maximum displacement, i.e., amplitude, and is given by \( A \omega^2 \), where \( \omega \) is the angular frequency.
02

Calculate the Frequencies

As both systems have the same mass \( m \) and total energy, the spring constants \( k \) must be the same to maintain equilibrium. Because the frequency \( f \) of a mass-spring system is related to the spring constant \( k \) and mass \( m \) by the formula \( f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \), it follows that both systems must have the same frequency, as \( f_1 = f_2 \).
03

Calculate the Maximum Accelerations

The maximum acceleration, \( a_{max} \), happens when the spring's potential energy is at its maximum - the extreme displacement or amplitude. Thus, \( a_{max} = A\omega^2 \), where \( A \) is the amplitude and \( \omega = 2\pi f \). For System 1, \( A_1 = 2A_2 \). So the maximum acceleration of System 1 would be \( a_{1_{max}} = 2A_2 \omega^2 = 2 a_{2_{max}} \) Thus, the maximum acceleration for System 1 is twice that of System 2 due to System 1's amplitude being twice that of System 2.

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