Question

if the earth were a homogenous sphere and a straight hole bored in it through its center . show that if a body were dropped into the hole , it would execute a simple hormonic motion. also find its time period .

Answer 1

FOLLOW THESE STEPS
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#01 : Find the equilibrium position of body (mean position) . PE is minimum and NET force is ZERO at that position.
We can find it at the center of the tunnel (centre of earth ) where gravity is ZERO and hence Force (=mg ) is ZERO.
IT IS THE EQUILIBRIUM POSITION.
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#02: displace the body from its mean position (say by small displacement 'r')
Now the force(F) acting on it :
F = m×g'
where m is the mass of the body and g' is the acceleration due to the gravity at that point .
we know that gravity at depth d (where our body is after being displaced ) is given by =
g[1-(d/R)]
= g[R-d]/R
here R is radius of Earth and g is gravity at surface of earth .
Now R-d = r .
=> g' = gr/R .
=> F = - mgr/R
This force tends to bring the body back to its mean position hence it will act always in opposite direction of the displacement (by us) . hence -ve sign in the above expression.
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#03 : We have to show that this force is proportional to displacement :
F= -mgr/R
=> F =- (mg/R)× r
=> F is proportional to displacement
It can be compared with SHM required condition . (F=-kx)
=> THE BODY EXECUTES SIMPLE HARMONIC MOTION**
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#04: Find acceleration (a) by dividing F by m :
a=- gr/R .
equating with [w²r] ;we get ;
gr/R = w²r
=> w = √(g/R)
here w is angular frequently .
we know that --> w = 2Π/T
where T is the time period of SHM.
we get;
2Π/T = √(g/R)
=> T = 2π√(R/g)
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Hope it helps!

Answer 2

Answer:

Let a body of mass 'm' be dropped in straight hole in the earth of mass M, density p and radius R. The body will be attracted towards the centre of the

earth with a force given by,

or

When the body is dropped into the straight hole, and it falls through the depth 'd' the value of acceleration due to gravity.

or

i.e., acceleration of the body is directly proportional to the displacement from the centre of earth O. Thus the motion is SHM.

{tex}T = 2\pi \sqrt {\frac{{Displacement}}{{Acceleration}}} \{/tex} Or

Or