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Consider the Earth as a homogenous sphere of radius R and a straight hole is bored in it through its centre. Show that a particle dropped into the holewill execute a simple harmonic motion such that its time period is T= 2pi sqrt((R)/(g))

Answer» <html><body><p></p><a href="https://interviewquestions.tuteehub.com/tag/solution-25781" style="font-weight:bold;" target="_blank" title="Click to know more about SOLUTION">SOLUTION</a> :`T=2pisqrt((R)/(g))` <br/> Oscillations of a particle dropped in a tunnel along the diameter of the earth. <br/> Consider earth to be a sphere of <a href="https://interviewquestions.tuteehub.com/tag/radius-1176229" style="font-weight:bold;" target="_blank" title="Click to know more about RADIUS">RADIUS</a> R and centre O. A straight tunnel is dug along the diameter of the earth. Let .g. be the value of acceleration due to gravity at the surface of the earth. <br/> Suppose a body of mass .m. is dropped into the tunnel and it is at point P. i.e., at a depth d below the surface of the earth at any instant. <br/> If g. is acceleration due to gravity at P. <br/> then `g.=g(1-(d)/(R))=g((R-d)/(R))` <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/FM_PHY_XI_V02_C10_E01_041_S01.png" width="80%"/> <br/> If y is distance of the body from the centre of the earth, then <br/> `R-d=y` <br/> `therefore g.=g((y)/(R))` <br/> Force acting on the body a point P is <br/> `F=-mg.=-(mg)/(R)y" i.e., "Fpropy` <br/> Negative sign indicates that the force <a href="https://interviewquestions.tuteehub.com/tag/acts-848461" style="font-weight:bold;" target="_blank" title="Click to know more about ACTS">ACTS</a> in the opposite direction of displacement. <br/> Thus the body will execute SHM with force <a href="https://interviewquestions.tuteehub.com/tag/constant-930172" style="font-weight:bold;" target="_blank" title="Click to know more about CONSTANT">CONSTANT</a>, `k=(mg)/(R)` <br/> The period of oscillation of the body will be `T=2pisqrt((m)/(k))=2pisqrt((m)/(mg//R))` <br/> `T=2pisqrt((R)/(g))`</body></html>


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