(D) The metallic wire acts as a spring with an effective spring constant $k_w = \frac{YA}{L}$.The wire and the spring are connected in series.The equi

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$2\pi \sqrt{\frac{m(KL + YA)}{KYA}}$

(D) The metallic wire acts as a spring with an effective spring constant $k_w = \frac{YA}{L}$.The wire and the spring are connected in series.The equivalent spring constant $k_{eq}$ for two springs in series is given by $\frac{1}{k_{eq}} = \frac{1}{k_w} + \frac{1}{K}$.Substituting $k_w$,we get $\frac{1}{k_{eq}} = \frac{L}{YA} + \frac{1}{K} = \frac{KL + YA}{KYA}$.Thus,$k_{eq} = \frac{KYA}{KL + YA}$.The time period $T$ of a mass-spring system is $T = 2\pi \sqrt{\frac{m}{k_{eq}}}$.Substituting $k_{eq}$,we get $T = 2\pi \sqrt{\frac{m(KL + YA)}{KYA}}$.

Class 11 Physics Oscillations SHM of Spring Mass System

Difficult

One end of a long metallic wire of length $L$,area of cross-section $A$,and Young's modulus $Y$ is tied to the ceiling. The other end is tied to a massless spring of force constant $K$. $A$ mass $m$ hangs freely from the free end of the spring. It is slightly pulled down and released. Its time period is given by

A
$2\pi \sqrt{\frac{m}{K}}$
B
$2\pi \sqrt{\frac{mYA}{KL}}$
C
$2\pi \sqrt{\frac{mK}{YA}}$
D
$2\pi \sqrt{\frac{m(KL + YA)}{KYA}}$

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SHM of Spring Mass System

$A$ block is placed on a frictionless horizontal table. The mass of the block is $m$ and springs are attached on either side with force constants $K_1$ and $K_2$. If the block is displaced a little and left to oscillate,then the angular frequency of oscillation will be

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Four massless springs whose force constants are $2K, 2K, K$ and $2K$ respectively are attached to a mass $M$ kept on a frictionless plane as shown in the figure. If mass $M$ is displaced in the horizontal direction,then the frequency of the oscillating system is:

DifficultMHT CET 2023

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$A$ sphere of mass $m$ is attached to the lower end of a light vertical spring with force constant $k$. The upper end of the spring is fixed. The sphere is released from rest at the natural (unstretched) length of the spring and comes to rest again after falling through a distance $x$.

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$A$ mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9m$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2}$ is . . . . . . .

DifficultJEE MAIN 2024

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$A$ force of $6.4 \ N$ stretches a vertical spring by $0.1 \ m$. The mass that must be suspended from the spring so that it oscillates with a time period of $\pi/4 \ s$ is .... $kg$.

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$A$ block is placed on a frictionless horizontal table. The mass of the block is $m$ and springs are attached on either side with force constants $K_1$ and $K_2$. If the block is displaced a little and left to oscillate,then the angular frequency of oscillation will be

Four massless springs whose force constants are $2K, 2K, K$ and $2K$ respectively are attached to a mass $M$ kept on a frictionless plane as shown in the figure. If mass $M$ is displaced in the horizontal direction,then the frequency of the oscillating system is:

$A$ sphere of mass $m$ is attached to the lower end of a light vertical spring with force constant $k$. The upper end of the spring is fixed. The sphere is released from rest at the natural (unstretched) length of the spring and comes to rest again after falling through a distance $x$.

$A$ mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9m$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2}$ is . . . . . . .

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