Question 108: Oscillations

Question

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :

Accepted Answer

Value of g on the particle of mass m,
$$g = {{GMd} \over {{R^3}}}$$
Force acting on the particle towards the center of the earth,
F = mg
Force along the tunnel = F cos$$	heta$$ = F
1
= mg cos$$	heta$$
= m . $${{GMd} \over {{R^3}}}$$$$\left( {{x \over d}} ight)$$
= $${{GMm} \over {{R^3}}}$$ . x
= $${{{g_s}m} \over R}.\,x$$ [as g
s
= $${{GM} \over {{R^2}}}$$ at earth surface]
$$ 	herefore $$ acceleration along the tunnel
$$\alpha$$ = $${{{F_1}} \over m}$$
= $${{{g_s}} \over R}.x$$
Time period = $${{2\pi {} } \over \omega } = 2\pi {\sqrt {{x \over \alpha }} } $$ [as $${\omega ^2} = {\alpha  \over x}$$]
$$ = 2\pi {\sqrt {{R \over {{g_s}}}} } $$

Answer

$$2\pi \sqrt {{R \over g}} $$

Answer

$${g \over {2\pi R}}$$

Answer

$${{2\pi R} \over g}$$

Answer

$${1 \over {2\pi }}\sqrt {{g \over R}} $$

Oscillations

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :

Choose the correct answer: