Consider the earth as a uniform sphere if mass `M` and radius `R`. Imagine a straight smooth tunnel made through the earth which connects any two points on its aurface. Show that the motion of a particle of mass m along this tunnel under the action of gravitation would be simple harmonic. Hence, determine the time that a particle would take to go from one end to the other through the tunnel.

Answer

C

Explanation

Suppose at some instant, the particle is at radial distance `r` from centre of earth `O`. Since, the particle is constrained to move along the tunnel, we define its position as distance `x` from `C`. Hence, equation of motion of the particle is,
`ma_(x) = F_(x)`
The gravitational force on mass `m` at distance `r` is,
`F = (GMmr)/(R^(3))` (towards `O`)
Therefore, `F_(x) = - F sin theta`
` = - (GMmr)/(R^(3))((x)/(r))`
`= - (GMm)/(R^(3)).x`
Since, `F_(x) prop - x`, motion is simple harmonic in nature. Further,
` ma_(x) = - (GMm)/(R^(3)). x`
or `a_(x) = - (GM)/(R^(3)).x`
`:.` Time period of oscillstion is, `T = 2pi sqrt(|(x)/(a_(x))|)`
` = 2pi sqrt((R^(3))/(GM))`
The time taken by particle to go from one end to the other is `(T)/(2)`.
`:. t = (T)/(2)`
`= pi sqrt((R^(3))/(GM))`
.