A small mass measuring device (SMMD) used for research on...

A small mass measuring device (SMMD) used for research on the biological effects of spaceflight consists of a small springmounted cage. Rats or other small subjects are introduced into the cage, which is set into oscillation. Calibration of a SMMD gives a linear function for the square of the oscillation period versus the subject's mass $m$ in $\mathrm{kg}: T^{2}=4.0 \mathrm{s}^{2}+\left(5.0 \mathrm{s}^{2} / \mathrm{kg}\right) \mathrm{m} .$ Find (a) the spring constant and (b) the mass of the cage alone.

Key Concepts

Spring Constant

The spring constant, denoted by k, measures the stiffness of a spring. It quantifies the force required to produce a unit displacement from equilibrium, and in the context of simple harmonic motion, it directly influences the oscillation period of the mass-spring system.

Oscillation Period-Mass Relationship

In mass-spring systems undergoing simple harmonic motion, the square of the oscillation period is directly proportional to the effective mass (which may include added components, such as the mass of a cage or mount). This relationship is often expressed as T² = (4?²/k)(m + m?), highlighting the linear dependence of T² on mass.

Effective Mass

The effective mass in a mass-spring system accounts for all the mass contributing to the inertia during oscillation. This includes not just the added test mass but also any inherent mass of the apparatus (such as the supporting cage), which shifts the system’s dynamic behavior and is reflected in the intercept of a calibration curve.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position. It serves as a fundamental model for systems such as mass-spring oscillators, describing how they move back and forth in a regular, sinusoidal pattern.

Mass-Spring System

A mass-spring system is a physical setup where a mass is attached to a spring, and its motion is governed by Hooke’s law. The dynamics of such systems are typically analyzed using the SHM framework, where the period of oscillation depends on both the mass of the object and the spring’s stiffness.

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