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Physics-

General

Easy

Question

Initially spring is compressed by distance $x subscript 0 end subscript$ from equilibrium position. At this compression block is given velocity $square root of fraction numerator 3 k over denominator m end fraction end root x subscript 0 end subscript comma$ so that compression in the spring increases and block start SHM (Spring constant K). Equation of motion of the block is

$y = 3 x_{0} \sin [\sqrt{\frac{k}{m}} t + \frac{π}{3}]$
$y = 2 x_{0} \sin [\sqrt{\frac{k}{m}} t + \frac{π}{6}]$
$y = 3 x_{0} \sin [\sqrt{\frac{k}{m}} t + \frac{π}{6}]$
$y = 2 x_{0} \sin [\sqrt{\frac{k}{m}} t + \frac{π}{3}]$

The correct answer is: $y equals 2 x subscript 0 end subscript sin invisible function application open square brackets square root of fraction numerator k over denominator m end fraction end root t plus fraction numerator pi over denominator 6 end fraction close square brackets$

From conservation of energy
$fraction numerator 1 over denominator 2 end fraction K x subscript 0 end subscript to the power of 2 end exponent plus fraction numerator 1 over denominator 2 end fraction m open square brackets square root of fraction numerator 3 K over denominator m end fraction end root x subscript 0 end subscript close square brackets to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction K x to the power of 2 end exponent$
$fraction numerator 1 over denominator 2 end fraction K x subscript 0 end subscript to the power of 2 end exponent plus fraction numerator 3 over denominator 2 end fraction K x subscript 0 end subscript to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction K x to the power of 2 end exponent$
$therefore x equals 2 x subscript 0 end subscript left parenthesis A m p l i t u d e right parenthesis$
Equation $y equals 2 x subscript 0 end subscript sin invisible function application open square brackets square root of fraction numerator K over denominator m end fraction end root t plus fraction numerator pi over denominator 6 end fraction close square brackets$ satisfies the given condition

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