1.

Which of the following statements is true ? (i) A car of mass M is tied by one end of a massless rope of length 10 m. The other end of the rope is in the hands of a man of mass M. The entire system is on a smooth horizontal surface.The man is at x = 0 and the cart at x = 10 m. If the man pulls the cart by the rope, the man and the cart will meet at the point x = 5 m. (ii) Two spherical bodies of mass M and 5 M and radii R and 2 R, respectively , are released in free space with initial separation between their centers equal to 12 R. If they attract each other due to the gravitational force only , then the distance covered by the smaller body just before collision is 7.5 R. (iii) Two skaters A and B , having masses 40 kg and 60 kg , respectively stand facing each other 10 m apart on a horizontal smooth surface. They pull on a rope stretched between them , the distance covered by A, when skaters meet is 6 m. (iv) A ballon (mass M) is attached to light rope of length L. To the other end of the rope a boy (mass m) is hanging in air. The system is at rest. The distance travelled by the balloon (in downward direction) when the boy touches the ballon is (ML)/(M + m).

Answer»

`(i),(ii)`
`(i) , (ii) ,(iii)`
`(iii) ,(iv)`
all of the above

Solution :Initially when the SYSTEM is at rest (i.e.`v_(c.m.) = 0`) and the net EXTERNAL force is zero , the location of center of mass is fixed. The particles will meet at center of mass is fixed. The particles will meet at center of mass (if the particles are treated as point masses).
(i) `Mx_(1) = Mx_(2) , x_(1) + x_(2) = L rArr x_(1) = x_(2) = (L)/(2) = 5 m`
(ii) `Mx = 5 M( 9 R - x) rArr x = 7.5 R`
`40 x = 60(10 - x) rArr x = 6 m`
(iv) `Mx = m(L - x) rArr x = (mL)/(M + m)`


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