A particle of mass m is dropped in a tunnel dug along the diameter of the earth, from the surface find. (1) Its energy of oscillation (2) Time taken to move from surface to a depth Re/2 (3) Velocities at a depth Re/2 and centre (4) graph between a Vs v (5) If it is released from a height h above the surface, will the motion be SHM ?

A particle of mass m is dropped in a tunnel dug along the diameter of

1.	A particle of mass m is dropped in a tunnel dug along the diameter of the earth, from the surface find. (1) Its energy of oscillation (2) Time taken to move from surface to a depth Re/2 (3) Velocities at a depth Re/2 and centre (4) graph between a Vs v (5) If it is released from a height h above the surface, will the motion be SHM ?
Answer» Solution :(1) ENERGY of oscillation `E = (1)/(2) mw^(2)A^(2) = (1)/(2)m((g)/(Re)) Re^(2) = (1)/(2) mg Re` (2) TIME taken to Re/2 x = A cos wt `(Re)/(2) = Re cos wt` `wt = (pi)/(3), (2pi)/(T)t = (pi)/(3)` `t = (T)/(6) = (2pi)/(6) sqrt((Re)/(g)) = (pi)/(3) sqrt((Re)/(g))` (3) `v = w sqrt(A^(2) x^(2))` `v = sqrt((g)/(Re)) sqrt(Re^(2) - (Re^(2))/(4)) = sqrt((g)/(RE)) xx Re (sqrt(3))/(2) = (sqrt(3g Re))/(2)` (4) `because "" v = w sqrt(A^(2) - x^(2))` `therefore""(v^(2))/(w^(2)A^(2)) + (x^(2))/(A^(2)) = 1` (5) Out side the earth.s SURFACE the FORCE is gravitational which is not proportional to .x.. Therefore the motion will not S.H.M. but oscillatory.