A 36-kg mass is placed on a horizontal frictionless surface...

A 36-kg mass is placed on a horizontal frictionless surface and then connected to walls by two springs with spring constants $k_{1}=3.0 \mathrm{~N} / \mathrm{m}$ and $k_{2}=4.0 \mathrm{~N} / \mathrm{m},$ as shown in the figure. What is the period of oscillation for the 36 -kg mass if it is displaced slightly to one side? a) $11 \mathrm{~s}$ b) $14 \mathrm{~s}$ c) $17 \mathrm{~s}$ d) $20 .$ s e) $32 \mathrm{~s}$ f) 38 s

A 36-kg mass is placed on a horizontal frictionless surface and then connected to walls by two springs with spring constants $k_{1}=3.0 \mathrm{~N} / \mathrm{m}$ and $k_{2}=4.0 \mathrm{~N} / \mathrm{m},$ as shown in the figure. What is the period of oscillation for the 36 -kg mass if it is displaced slightly to one side?
a) $11 \mathrm{~s}$
b) $14 \mathrm{~s}$
c) $17 \mathrm{~s}$
d) $20 .$ s
e) $32 \mathrm{~s}$
f) 38 s

Key Concepts

Mass-Spring System

A mass-spring system is a classical model used to study oscillatory motion where a mass is attached to a spring that obeys Hooke's Law. When the mass is displaced from its equilibrium position, the restoring force, which is proportional to the displacement, causes it to oscillate around the equilibrium. This system is central to understanding dynamics in mechanical and vibrational analysis.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion in which the net force acting on the system is directly proportional to the displacement from its equilibrium and is directed towards that equilibrium. The motion is characterized by its sinusoidal time dependence and consistent period for small amplitude oscillations, making it a fundamental concept in oscillatory systems.

Effective Spring Constant

When two or more springs are attached to a mass, their combined effect can be represented by an effective spring constant. For springs arranged in parallel, as is often the case when connected to fixed supports, the effective spring constant is the sum of the individual spring constants. This effective constant simplifies the analysis of the system's dynamics by allowing the use of the standard SHM formulas.

Period of Oscillation

The period of oscillation is the time required for an oscillating system to complete one full cycle. In the context of a mass-spring system undergoing simple harmonic motion, the period is calculated using the formula T = 2??(m/k), where m represents the mass of the object and k is the effective spring constant. This period depends on the balance between the inertial mass and the stiffness of the spring system.

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