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Question

# Two uniform solid spheres of equal radii R, but mass M and 4M have a centre to centre separation 6R, as shown in figure. A projectile of mass m is projected from the surface of the sphere of mass M directly towards the centre of the second sphere. The minimum speed of the projectile so that it reaches the surface of the second sphere is

A
45GMR
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B
54GMR
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C
35GMR
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D
53GMR
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Solution

## The correct option is B √35GMRLe the projectile of mass m is fired with minimum velocity, v from the surface of the sphere of mass M to reach the surface of the sphere of mass 4M. Let N be a neutral point at a distance r from the center of the sphere of mass M. At neutral point N, GMr2=G(4M)m(6R−r)2(6R−r)2=4r26R−r=±2r or r=2R or −6RThe point r=−6R does not concern us.Thus, ON=r=2RIt is sufficient to project the projectile with a speed that would enable it to reach N. Thereafter, the greater gravitational pull of 4M would suffice.The mechanical energy at the surface of M is Ei=12mv2−GMmR−G(4M)m5RAt the neutral point N, speed approaches zero.∴ The mechanical energy at N is EN=−GMm2RG(4M)m4R=−GMm2R−GMmRAccording to law of conservation of mechanical energy, Ei−EN12mv2−GMmRG(4M)m5R=−GMm2R−GMmRv2=2GMR[45−12]=35GMR or v=(35GMR)1/2

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