As shown in figure, two light springs having force constants k1=1.8 N m^-1 and k2=3.2 N m^-1

step 1 Hello and welcome to this video solution of Numerate. Here we are given a spring mass system so

As shown in figure, two light springs having force constants...

As shown in figure, two light springs having force constants $\mathrm{k}_{1}=1.8 \mathrm{~N} \mathrm{~m}^{-1}$ and $\mathrm{k}_{2}=3.2 \mathrm{~N} \mathrm{~m}^{-1}$ and a block having mass $\mathrm{m}=200 \mathrm{~g}$ are placed on a frictionless horizontal surface. One end of both springs are attached to rigid supports. The distance between the free ends of the spring is $60 \mathrm{~cm}$ and the block is moving in this gap with a speed $\mathrm{v}=120 \mathrm{~cm} \mathrm{~s}^{-1}$.When the block is moving towards $k_{1}$, what will be the time taken for it to get maximum compressed from point $\mathrm{C}$ ? (A) $\pi \mathrm{s}$ (B) $(2 / 3) \mathrm{s}$ (C) $(\pi / 3) \mathrm{s}$ (D) $(\pi / 4) \mathrm{s}$

As shown in figure, two light springs having force constants $\mathrm{k}_{1}=1.8 \mathrm{~N} \mathrm{~m}^{-1}$ and $\mathrm{k}_{2}=3.2 \mathrm{~N} \mathrm{~m}^{-1}$ and a block having mass
$\mathrm{m}=200 \mathrm{~g}$ are placed on a frictionless horizontal surface. One end of both springs are attached to rigid supports. The distance between the free ends of the spring is $60 \mathrm{~cm}$ and the block is moving in this gap with a speed $\mathrm{v}=120 \mathrm{~cm} \mathrm{~s}^{-1}$.When the block is moving towards $k_{1}$, what will be the time taken for it to get maximum compressed from point $\mathrm{C}$ ?
(A) $\pi \mathrm{s}$
(B) $(2 / 3) \mathrm{s}$
(C) $(\pi / 3) \mathrm{s}$
(D) $(\pi / 4) \mathrm{s}$

Key Concepts

Period of Oscillation and Phase Relationships

The period of oscillation in a spring?mass system, given by T = 2??(m/k_eff), is the time required to complete one full cycle of motion. Understanding that specific points in the oscillatory motion correspond to specific fractions of the period—such as a quarter period representing the time taken to go from the equilibrium position to maximum displacement—is crucial in analyzing timing problems in SHM.

Effective Spring Constant

When multiple springs interact with a single mass, the system behaves as if acted upon by a single spring with an effective spring constant. In configurations where springs work together (either in series or parallel), their individual force constants combine in a way that determines the overall stiffness of the system, which in turn influences the oscillation frequency and period.

Hooke's Law

Hooke's Law states that the restoring force exerted by a spring is directly proportional to the displacement from its equilibrium position, acting in the opposite direction of the displacement. This principle is fundamental in analyzing any spring system and in determining how the force constant of the spring governs its response to compression or extension.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes systems where the net restoring force is proportional to the displacement and directed towards the equilibrium. This results in sinusoidal oscillatory motion, where quantities like displacement, velocity, and acceleration vary periodically with time, characterized by an angular frequency that depends on the mass and the effective spring constant.

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