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Q.

A ring of radius R = 4 m made of a highly dense material has a mass m1=5.4×109kg distributed uniformly over its circumference. A highly dense particle of mass m2=6×108kg is placed on the axis of the ring at a distance 3 m from the center of the ring. Find the speed of the particle (in cm/s), when the particle is at the center of the ring. Except mutual gravitational interaction of the two, neglect all other forces.

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answer is 18.

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Detailed Solution

When the particle is at the center, let their respective speedsare v1 and v2.From conservation of linear momentum, 0=5.4×109v1−6×108v2∴ v2=9v1From conservation of energy, (U+K)i=(U+K)f−Gm1m242+32+0=−Gm1m24+12m1v12+12m2v22Gm1m214−15=12m1v12+12m281v12G9m2220=1290m2v12∵m1=9 m2 Now m1=9m2 (given) v1=Gm2100=6.67×10−11×6×108100=0.02ms−1∴ v2=9v1=0.18ms−1=18cms−1
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A ring of radius R = 4 m made of a highly dense material has a mass m1=5.4×109kg distributed uniformly over its circumference. A highly dense particle of mass m2=6×108kg is placed on the axis of the ring at a distance 3 m from the center of the ring. Find the speed of the particle (in cm/s), when the particle is at the center of the ring. Except mutual gravitational interaction of the two, neglect all other forces.