A tunnel is dug through the centre of the Earth. Show that a body of m

1.	A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from one end of the tunnel will execute simple harmonic motion.
Answer» Acceleration due to gravity inside Earth g′ = \(\frac{GM}{R^3}x=\frac{g}{R}x\) Here x = distance of the point from centre of earth (x<R) If, block of mass m is placed along the diameter inside the earth. So, force on block F = − \(\frac{mg}{R}x\) = −kx ∴ T = 2π\(\sqrt{\frac{m}{k}}\) = 2π\(\sqrt{\frac{m}{\frac{mg}{R}}}\) T = 2π\(\sqrt{\frac{R}{g}}\)

A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from one end of the tunnel will execute simple harmonic motion.

Answer»

Acceleration due to gravity inside Earth

g′ = \(\frac{GM}{R^3}x=\frac{g}{R}x\)

Here x = distance of the point from centre of earth (x<R)

If, block of mass m is placed along the diameter inside the earth.

So, force on block

F = − \(\frac{mg}{R}x\) = −kx

∴ T = 2π\(\sqrt{\frac{m}{k}}\) = 2π\(\sqrt{\frac{m}{\frac{mg}{R}}}\)

T = 2π\(\sqrt{\frac{R}{g}}\)

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A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from one end of the tunnel will execute simple harmonic motion.

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