1.

A small particle is dropped from a height R in front of a narow tunnel dug inside the earth (along a diameter). Let M be the mass of earth, R be radius of earrth. Let v_(0), T be speed of particle when it reaches A and time taken by particle to go from A to B respectively. Assuming mass of particle to be negligible as compare to mass of earth, pick the correct option(s)

Answer»

`V_(0)=SQRT((GM)/(2R))`
`V_(0)=sqrt((GM)/R)`
`R=(pi)/2sqrt((R^(3))/(GM))`
`T=pisqrt((R^(3))/(GM))`

Solution :`(-GMm)/(2R)=(-GMm)/R+1/2mv_(0)^(2)`
`impliesv_(0) =sqrt((GM)/R)`
while in the tunnel,
`(d^(2)X)/(dt^(2))=-((GM)/(R^(3)))x`
`v_(0)^(2)=(GM)/R=omega^(2)(a^(2)-R^(2))impliesa=sqrt(2)R`
`implies T_(ATOB)=1/4xx2pisqrt((R^(3))/(GM))`


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