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Question

A small sphere of mass 200 gm is attached to an inextensible string of length 130 cm whose upper end is fixed to the ceiling. The sphere is made to describe a horizontal circle of radius 50 cm. Calculate the time period of this conical pendulum and the tension in the string. (π2=10)

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Solution

Balancing the forces:

Tcosθ=mg(1)Tsinθ=mv2r(2)

Dividing (2) by (1)

tanθ=v2rg(sinθ=rl=50130θ=sin1(513)=22.62°)v2=rgtanθv=rgtanθ=50×10×0.42×102

=2.1=145m/s

Time period T=2πω=2πrvT=2πrv=2π×50×1021.45=2.16s

To calculate tension:

T2cos2θ+T2sin2θ=(mg)2+(mv2r)2T2=(mg)2+(mv2r)2T=(mg)2+(mv2r)2=(0.2×10)2+(0.2×2.1050×102)2=4+0.70=4.7=2.17N


1036548_1106731_ans_45ba5aa0d384471888cda2d9b60571a3.png

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