A mass m is suspended from two springs of spring constant k1...

A mass $m$ is suspended from two springs of spring constant $k_{1}$ and $k_{2}$ as shown in Fig. 9.23. The time period of vertical oscillations of the mass will be (A) $2 \pi \sqrt{\left(\frac{k_{1}+k_{2}}{m}\right)}$ (B) $2 \pi \sqrt{\frac{m}{\left(k_{1}+k_{2}\right)}}$ (C) $2 \pi \sqrt{\frac{m\left(k_{1} k_{2}\right)}{\left(k_{1}+k_{2}\right)}}$ (D) $2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{\left(k_{1} k_{2}\right)}}$

A mass $m$ is suspended from two springs of spring constant $k_{1}$ and $k_{2}$ as shown in Fig. 9.23. The time period of vertical oscillations of the mass will be
(A) $2 \pi \sqrt{\left(\frac{k_{1}+k_{2}}{m}\right)}$
(B) $2 \pi \sqrt{\frac{m}{\left(k_{1}+k_{2}\right)}}$
(C) $2 \pi \sqrt{\frac{m\left(k_{1} k_{2}\right)}{\left(k_{1}+k_{2}\right)}}$
(D) $2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{\left(k_{1} k_{2}\right)}}$

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Key Concepts

Oscillation Period of a Mass-Spring System

The oscillation period in a mass-spring system undergoing SHM is determined by the mass and the effective spring constant of the system. It is given by the formula T = 2??(m/Keffective), where Keffective takes into account the configuration and combination of springs used in the system.

Effective Spring Constant for Parallel Springs

When multiple springs are arranged in parallel, the effective spring constant is the sum of the individual spring constants. This means that the combined stiffness of the system increases, leading to a stronger restoring force for a given displacement compared to a single spring.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes oscillatory motion where the restoring force is directly proportional to displacement and acts in the direction opposite to that displacement. This principle underlies many mechanical systems such as oscillating springs, where the motion is periodic and can be mathematically described by sinusoidal functions.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position, expressed as F = -kx. This fundamental relationship is key to understanding the dynamics of systems involving springs, especially when determining the restoring force that leads to SHM.

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