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Physics-

General

Easy

Question

Two identical balls $A$ and $B$ each of mass $0.1 blank k g$ are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid smooth pipe bent in the form of circle as shown in the figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius $0.06 blank m.$ Each spring has a natural length of $0.06 pi blank m$ and force constant $0.1 N divided by m.$ Initially both the balls are displaced by an angle $theta equals pi divided by 6$ radian with respect to the diameter $P Q$ of the circle and released from rest. The frequency of oscillation of the ball $B$ is

$π H z$
$\frac{1}{π} H z$
$2 π H z$
$\frac{1}{2 π} H z$

The correct answer is: $fraction numerator 1 over denominator pi end fraction H z$

As here two masses are connected by two springs, this problem is equivalent to the oscillation of a reduced mass $m subscript r end subscript$ of a spring of effective spring constant
$T equals 2 pi square root of fraction numerator m subscript r end subscript over denominator K subscript e f f. end subscript end fraction end root$
Here $m subscript r end subscript equals fraction numerator m subscript 1 end subscript m subscript 2 end subscript over denominator m subscript 1 end subscript plus m subscript 2 end subscript end fraction equals fraction numerator m over denominator 2 end fraction rightwards double arrow K subscript e f f. end subscript equals K subscript 1 end subscript plus K subscript 2 end subscript equals 2 K$
$therefore n equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator K subscript e f f. end subscript over denominator m subscript r end subscript end fraction end root equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 2 K over denominator m end fraction cross times 2 end root equals fraction numerator 1 over denominator pi end fraction square root of fraction numerator K over denominator m end fraction end root equals fraction numerator 1 over denominator pi end fraction square root of fraction numerator 0.1 over denominator 0.1 end fraction end root equals fraction numerator 1 over denominator pi end fraction H z$

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