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Chapter 13: Problem 71

Two mass-spring systems with the same mass are undergoing oscillatory motion with the same amplitudes. System 1 has twice the frequency of system \(2 .\) How do (a) their energies and (b) their maximum accelerations compare?

Short Answer

Expert verified
The energy of system 1 is four times that of system 2, and the maximum acceleration of system 1 is twice that of system 2.

Step by step solution

01

Understand Energy in Harmonic Oscillations

The energy in a spring system undergoing harmonic motion can be described by \(E = \frac{1}{2}m蠅^2A^2\), where \(蠅\) is the angular frequency, \(A\) is the amplitude and \(m\) is the mass. Since the mass and amplitude are same for both systems, comparing energy comes down to comparing the square of their frequencies.
02

Compare the Energies

Since the frequency of system 1 is twice of system 2, when we square their frequencies, \((2蠅)^2 = 4蠅^2\) where \(蠅\) is the frequency of system 2. Thus, the energy of system 1 is four times the energy of system 2.
03

Understand Maximum Acceleration in Harmonic Oscillations

The maximum acceleration in this system can be described by \(a_{max} = 蠅^2A\), where \(蠅\) is the angular frequency and \(A\) is the amplitude.
04

Compare the Maximum Accelerations

Similarly as in Step 2, the maximum acceleration of system 1 is double that of system 2, since the angular frequency for system 1 is twice that of system 2, and the amplitudes are the same.

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