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?

In understanding the behavior of a mass-spring system, the frequency plays a critical role. The frequency of a mass-spring system is determined by its mass (m) and the spring constant (k), independent of the amplitude of oscillation. It is calculated using the formula:

\( f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \)

This intrinsic property of the system, called the natural frequency, dictates how often the mass completes a cycle of movement back and forth in a given time period. For two systems with identical mass and spring constant, the frequency remains constant regardless of the differences in amplitude. The amplitude reflects how far the mass moves from its equilibrium position, but it does not affect how quickly the system oscillates. This counterintuitive fact often surprises students who might initially think a larger amplitude would result in a slower frequency due to the greater distance covered. However, in physics, the defining factors for frequency in a mass-spring system are solely the mass and the spring constant.

When dealing with harmonic motion, such as that exhibited by a mass-spring system, the concept of maximum acceleration is intimately tied to the system's amplitude and angular frequency. The maximum acceleration occurs at the maximum displacement, which is the amplitude itself.
The equation capturing this relationship is given by:
\( A = \omega^2 \times \text{Amplitude} \)

Where \(A\) is the maximum acceleration, \(\omega\) is the angular frequency, and Amplitude is the maximum distance from the equilibrium position. Given that angular frequency is calculated as \(\omega = 2\pi\times f\), where \(f\) is the frequency, and considering that the frequency does not depend on amplitude, it follows that for two systems with the same mass, spring constant, and total energy, the system with the larger amplitude will exhibit greater maximum acceleration. Thus, although their oscillation rates - frequencies - are equivalent, the system with the larger amplitude reaches higher speeds and thus experiences greater acceleration at the points of maximum displacement.

Angular frequency, represented by \(\omega\), is a measure of how fast the mass-spring system undergoes oscillations. It is related to, but distinct from, the regular frequency (f). The angular frequency is vital in understanding various aspects of oscillations as it relates to both the velocity and the acceleration of the object in motion.
The relationship between angular frequency and the standard frequency is given by the formula:
\( \omega = 2\pi f \)
\(\omega\) represents the rate at which the system moves through its cycle in radians per second, making it a fundamental part of equations dealing with the motion of the oscillating system. Angular frequency is particularly noteworthy in the context of acceleration, as seen in the maximum acceleration formula. A higher angular frequency means a system can undergo more oscillations in the same period, leading to greater maximum velocity and, hence, greater maximum acceleration at the peak amplitude. Because angular frequency relates to the stiffness of the spring and the mass of the system, two systems with the same mass and spring constant, regardless of their amplitude, will have the same angular frequency and, consequently, the same rate of oscillation through their cycles.