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

Question

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

Infinity Learn · Accepted Answer

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.$$T=2$$$$π$$mrKeffHere $$mr=m1m2m1+m2=m2$$⇒keff.$$=k1+k2=2k$$∴$$n=12$$$$π$$$$keffmr=12$$$$π$$$$2km$$×$$2=1$$$$π$$$$km=1$$$$π$$$$0.10.1=1$$$$π$$Hz

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