A tunnel is dug in the earth across one of its diameter. two masses m and 2m are dropped from the two ends of the tunnel. The masses collide and stick to each other. they perform SHM, the amplitude of which is -----------------

rajan jha Grade: 12

        A tunnel is dug in the earth across one of its diameter. two masses m and 2m are dropped from the two ends of the tunnel. The masses collide and stick to each other. they perform SHM, the amplitude of which is -----------------

8 years ago

Answers : (1)

vikas askiitian expert

509 Points

							gravitational potential at center  = -3/2 (GM_e/R_e)
gravitational potential at surface = -GMe_/R_e
accleration of each particle = -GMx/R_e³ , this is independent of mass so both of these particle will meet at center of earth...
applying energy conservation for mass m
 mV²/2 - 3/2(GM_em/R_e) = -GM_em/R_e 
here in the above eq mass cancels out so velocity is independent of mass of particle
 V_m = V_2m = V = (GM_e/R_e)^1/2 _{................1}
now applying momentam conservation at center of earth
-mV + 2mV = (m+2m)V¹                           (due to opposite direction one of the velocity is -ve)
  3mV¹ = mV
      V¹ = V/3        ...............2   
now this combined mass will perform SHM because force acting on this mass is -GM_e3mx/R_e³
    F=3m(a)
   a (accleration)= -GM_ex/R_e³
     -W² x = -GM_ex/R_e³                                  (a = -w²x)
      W = (GM_e/R_e³)^1/2                .........3
now maximum velocity of particle is at mean position which is given by eq 2 ,
 V_max = V/3
 V_max = AW                             (from eq of shm)
    AW = V/3
    A = V/3W                           (A is amplitude of SHM)
from eq3 & eq1
  A = R_e/3
thus ampitude will be 1/3rd times radius of earth