Question

A particle $' P'$ of mass $m$ is placed on a smooth horizontal surface is attached to three identical springs $A, B$ and $C$ each of force constant $K$ as shown in figure. If the particle is pushed slightly against spring $A$ and released. Find the time period of oscillation of the particle.

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Accepted Answer

Let us apply a force $F$ at $' P'$ along positive $x$ -direction so that the system remains in equilibrium. This will compress the spring $A$ by $C$ and stretch the springs $B$ and $C$ by $' x_{1}' .$

In $Δ M O P : X^{2} + Y^{2} = l^{2}$

Differentiating above equation both sides, we get

$2 X d X + 2 Y d Y = 2 l \cdot d l \dots (i)$

As $y$ is constant hence $d Y = 0$

Now Eq. (i) becomes $2 X d X = 2 l d l$

or $\frac{d l}{d X} = \frac{x_{1}}{x_{0}} = \frac{X}{l} \Rightarrow x_{1} = x_{0} \cos θ \dots (ii)$

As the block is at equilibrium, then $F = K x_{0} + 2 K x_{1} \cos θ \dots (iii)$

From (ii) and (iii) we get, $F = K x_{0} + 2 K x_{0} \cos^{2} θ$

or $\frac{F}{x_{0}} = K (1 + 2 \cos^{2} θ) = K_{eff}$

It means, $ω^{2} = \frac{K_{eff}}{m} = \frac{K (1 + 2 \cos^{2} θ)}{m}$

$\Rightarrow T = 2 π \sqrt{\frac{m}{K (1 + 2 \cos^{2} θ)}}$

B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Linear and Angular Simple Harmonic Motion, Exercise 1: ILLUSTRATIONS with Hints & Solutions

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