A mass m is attached to a spring with spring constant k, as...

A mass $m$ is attached to a spring with spring constant $k$, as shown in Figure $2.53 .$ The mass is pulled to the right a distance of $0.2 \mathrm{m}$ and released. Rank the following spring-mass combinations according to their oscillation periods from shortest to longest. If any combinations have the same period, give them the same rank. You should assume that there is no friction between the mass and the horizontal surface. (a) $k=0.5 \mathrm{N} / \mathrm{m} ; m=0.25 \mathrm{kg}$ (b) $k=0.5 \mathrm{N} / \mathrm{m} ; m=0.50 \mathrm{kg}$ (c) $k=0.5 \mathrm{N} / \mathrm{m} ; m=1.00 \mathrm{kg}$ (d) $k=1.0 \mathrm{N} / \mathrm{m} ; m=0.25 \mathrm{kg}$ (e) $k=1.0 \mathrm{N} / \mathrm{m} ; m=0.50 \mathrm{kg}$

A mass $m$ is attached to a spring with spring constant $k$, as shown in Figure $2.53 .$ The mass is pulled to the right a distance of $0.2 \mathrm{m}$ and released. Rank the following spring-mass combinations according to their oscillation periods from shortest to longest. If any combinations have the same period, give them the same rank. You should assume that there is no friction between the mass and the horizontal surface.
(a) $k=0.5 \mathrm{N} / \mathrm{m} ; m=0.25 \mathrm{kg}$
(b) $k=0.5 \mathrm{N} / \mathrm{m} ; m=0.50 \mathrm{kg}$
(c) $k=0.5 \mathrm{N} / \mathrm{m} ; m=1.00 \mathrm{kg}$
(d) $k=1.0 \mathrm{N} / \mathrm{m} ; m=0.25 \mathrm{kg}$
(e) $k=1.0 \mathrm{N} / \mathrm{m} ; m=0.50 \mathrm{kg}$

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This concept underpins the behavior of many physical systems, including spring-mass systems.

Mass-Spring System

A mass-spring system consists of a mass attached to a spring that obeys Hooke's law, which states that the force exerted by the spring is proportional to the displacement from equilibrium. These systems provide a classic example of simple harmonic motion when friction is negligible.

Oscillation Period

The oscillation period is the time it takes for one complete cycle of motion in an oscillating system. In a mass-spring system, the period is given by the formula T = 2??(m/k), which explicitly shows how the period depends on the mass attached to the spring and the spring's stiffness (spring constant).

Dependence on System Parameters

The formula T = 2??(m/k) reveals that the oscillation period increases with a larger mass and decreases with a larger spring constant. Understanding this dependency is crucial for analyzing and comparing the dynamic responses of different spring-mass combinations.

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