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Q.

Two identical balls A and B each of mass 0.1 kg 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 a 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 m. Each springhas a natural length of 0.06m and force constant 0.1N/m. Initially both the balls are displaced by an angle θ=π/6 radian with respect to the diameter PQ of the circle andreleased from rest. The frequency of oscillation of the ball B is

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a

πHz

b

1πHz

c

2πHz

d

12πHz

answer is B.

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Detailed Solution

As here two masses are connected by two springs, this problem is equivalent to the oscillation of a reduced mass m, of a spring of effective spring constant.T=2πmrKeffHere mr=m1m2m1+m2=m2⇒keff.=k1+k2=2k∴n=12πkeffmr=12π2km×2=1πkm=1π0.10.1=1πHz
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Two identical balls A and B each of mass 0.1 kg 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 a 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 m. Each springhas a natural length of 0.06m and force constant 0.1N/m. Initially both the balls are displaced by an angle θ=π/6 radian with respect to the diameter PQ of the circle andreleased from rest. The frequency of oscillation of the ball B is