As shown in figure, a body having mass m is attached with two springs having spring constants k1

step 1 Hello and welcome to this video solution of numerate. So we have given a figure which is there a

As shown in figure, a body having mass m is attached with...

As shown in figure, a body having mass $\mathrm{m}$ is attached with two springs having spring constants $\mathrm{k}_{1}$ and $\mathrm{k}_{2}$. The frequency of oscillation is $\mathrm{f}$. Now, if the springs constants of both the springs are increased 4 times, then the frequency of oscillation will be equal to .......... (A) $2 \mathrm{f}$ (B) $\mathrm{f} / 2$ (C) $\mathrm{f} / 4$ (D) $4 \mathrm{f}$

As shown in figure, a body having mass $\mathrm{m}$ is attached with two springs having spring constants $\mathrm{k}_{1}$ and $\mathrm{k}_{2}$. The frequency of oscillation is $\mathrm{f}$. Now, if the springs constants of both the springs are increased 4 times, then the frequency of oscillation will be equal to ..........
(A) $2 \mathrm{f}$
(B) $\mathrm{f} / 2$
(C) $\mathrm{f} / 4$
(D) $4 \mathrm{f}$

Key Concepts

Simple Harmonic Motion

This concept describes how a system oscillates about its equilibrium position when the restoring force is proportional to the displacement, as in many oscillatory systems. The relationship between the force and displacement (often given by Hooke's law) results in periodic motion characterized by amplitude, frequency, and phase.

Mass-Spring Oscillator Frequency

A mass attached to a spring exhibits oscillatory behavior with a frequency that is determined by the stiffness of the spring (spring constant) and the mass. The oscillation frequency is described by the formula f = (1/(2?))?(k/m), making it clear that any change in the spring constant or the mass will impact the frequency of oscillation.

Effective Spring Constant

When a system uses more than one spring, the overall or effective spring constant depends on how the springs are arranged (in series or parallel). For springs arranged in parallel, the effective spring constant is the sum of the individual constants, and for series arrangements, the reciprocal of the effective constant is the sum of the reciprocals of the individual constants. Understanding this concept is key to analyzing more complex oscillatory systems.

Scaling Effects in Oscillation Frequency

This concept addresses how the frequency of oscillation changes when the spring constant is modified. Since the frequency is proportional to the square root of the spring constant, if the spring constant is increased by a factor, the frequency increases by the square root of that factor. For example, quadrupling the spring constant results in a doubling of the frequency.

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