Imagine we drill a small tunnel that goes straight down from the North Pole to t

Question

Imagine we drill a small tunnel that goes straight down from the North Pole to the south pole of Earth. An astronaut wears a protective suit and goes into the tunnel and drops down. Due to the force of gravity, the astronaut gets trapped inside the tunnel, moving up and down between the North Pole and the South Pole, stopping at the North Pole only to travel back down to the South Pole, speeding up at the center of Earth. Find a numerical value for the time it takes the astronaut to travel from the north pole to the south pole, assuming the density of Earth is the same throughout. Mass of Earth: 5.972 times 10^24kg. Radius of Earth: 6371km Gravitational Constant: 6.67 times 10^-11Nm^2/kg^2 Mass of astronaut with gear: 120kg.

Explanation / Answer

at height h from centre of earth.

mass of earth below astronaut M' = volume x density

= M (4 pi r^3 / 3) / (4 pi R^3 / 3 )

= M (r^3 / R^3 )

Gravitational force , Fg = G M'm / r^2

= G (M (r^3 / R^3 ) ) m / r^2

= (G M m / R^3 ) r

here F = k r

so he will perform SHM.

and time period of SHM = 2 pi sqrt(m /k )

T = 2 pi sqrt( R^3 / G M)

T = 2pi sqrt[ (6371 x 10^3)^3 / (6.67 x 10^-11 x 5.972 x 10^24)]

T = 5062.54 sec

time taken to go from north to south pole = T/2 =2531.27 sec

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