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Physics-

General

Easy

Question

A mass M is attached to a horizontal spring of force constant k fixed on one side to a rigid support as shown in figure. The mass oscillates on a frictionless surface with time period T and amplitude A . When the mass is in equilibrium position. Another mass m is gently placed on it. What will be the new amplitude of oscillations?

$A \sqrt{(\frac{M}{M - m})}$
$A \sqrt{(\frac{M - m}{M})}$
$A \sqrt{(\frac{M}{M + m})}$
$A \sqrt{(\frac{M + m}{M})}$

The correct answer is: $A square root of open parentheses fraction numerator M over denominator M plus m end fraction close parentheses end root$

When a mass $m$ is placed on mass $M$ , the new system is of mass $equals left parenthesis M plus m right parenthesis$ , attached to the spring. New time period of oscillation,
$T to the power of ´ end exponent equals 2 pi square root of fraction numerator M plus m over denominator k end fraction end root$
$T equals 2 pi square root of fraction numerator M over denominator k end fraction end root$
Let $v equals$ velocity of the mass $M$ while passing through the mean position.
$v to the power of ´ end exponent equals$ Velocity of the mass $left parenthesis M plus m right parenthesis$ , while passing through the mean position.
According to law of conservation of linear momentum $M v equals open parentheses M plus m close parentheses v to the power of ´ end exponent$
At mean position, $v equals A blank omega$ and $v to the power of ´ end exponent equals A to the power of ´ end exponent omega to the power of ´ end exponent$
$therefore blank M A omega equals open parentheses m plus m close parentheses A to the power of ´ end exponent omega to the power of blank$
or $A to the power of ´ end exponent equals open parentheses fraction numerator M over denominator M plus m end fraction close parentheses fraction numerator omega over denominator omega end fraction comma A equals fraction numerator M over denominator M plus m end fraction cross times fraction numerator T to the power of ´ end exponent over denominator T end fraction cross times A$
$equals open parentheses fraction numerator M over denominator M plus m end fraction close parentheses cross times square root of fraction numerator M plus m over denominator M end fraction end root cross times A$
$equals A square root of fraction numerator M over denominator M plus m end fraction end root$

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