Question

A tunnel is dug to the centre of the earth. Show that a body of mass 'm' when dropped from rest from one end of the tunnel will execute simple harmonic motion.​

Answer 1

Answer:

Time period of SHM when a body is dropped from one end of the tunnel is given as

$T = 2\pi \sqrt{\frac{R^3}{GM}}$

Explanation:

As we know that the gravity inside the Earth at a distance "x" from the center of Earth is given as

$g = \frac{GMx}{R^3}$

now the force on the point mass inside the tunnel due to gravity is given as

$F = \frac{GMmx}{R^3}$

now the acceleration of the point mass is given as

$a = \frac{F}{m}$

$a = \frac{GM}{R^3} x$

since this force is always towards the center of Earth so we have

$\omega^2 = \frac{GM}{R^3}$

so the object will execute SHM with time period

$T = 2\pi \sqrt{\frac{R^3}{GM}}$

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Topic : SHM

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Answer 2

Answer:

Time period of SHM when a body is dropped from one end of the tunnel is given as

$</p><p>T = 2\pi \sqrt{\frac{R^3}{GM}}</p><p>$

Explanation:

As we know that the gravity inside the Earth at a distance "x" from the center of Earth is given as

$g = \frac{GMx}{R^3}$

now the force on the point mass inside the tunnel due to gravity is given as

$F = \frac{GMmx}{R^3}$

now the acceleration of the point mass is given as

$a = \frac{F}{m}$

$a = \frac{GM}{R^3}$

since this force is always towards the center of Earth so we have

$\omega^2 = \frac{GM}{R^3}$

so the object will execute SHM with time period

$T = 2\pi \sqrt{\frac{R^3}{GM}}T=2π </p><p>$

#Learn

Topic : SHM

https://brainly.in/question/7909806