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.

Question

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.

Infinity Learn · Accepted Answer

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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