Time period in different cases

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In this how they hwve written $T_1$
for simple pendulum I know $T=2\pi \sqrt\frac{l}{g}$

I also could not understand how they have written $F_2$

1st case:

The complete formula for time period of simple pendulum is $T = 2\pi \sqrt {\frac {1}{g (1/L + 1/R)}}$
Its derivation can be seen here

But when L << R then the formula reduces to $T=2\pi \sqrt {L/g}$

2nd case :
you may draw a circle as earth take center O . Make a tunnel along diameter. Let a body fall along tunnel and has displacement y. As the body approaches center the force decreases because here we know force acts on a particle at a distance y from its center is only due to the part of earth inside and outer part does not exert any net force.
Hence $F= -GM' m/ y^2$ -----1

M= total mass of earth. M' = effective mass of earth.

$M' = \rho 4/3 \pi y^3$

so $M'/M = y^3/R^3$

substitute M' in equation 1 .

$F=-( GmM/R^3 )×y$

Compare it with$F=-kx$ take out$k$ and finally put it into:

$2\pi \sqrt {m/k }$
You will get $T=2\pi \sqrt {R/g}$.

3rd case
$mw^2R = mg$ on solving you would get :
$T=2\pi \sqrt {R/g}$.

update : m is mass of pendulum.

answered Mar 25, 2017 by (2,320 points)
selected Mar 26, 2017 by koolman
From this "the part of earth inside and outer part "
What do you mean by outer and inner part . And 'part' of what ?
lets take example from an analogy of electrostatics, when we applied gauss law on a non conducting sphere to take out foeld at an "inside point" i.e. less than its radius we take only the field due to charge only inside same is happening here .
Case 1: Where does the "complete formula" come from? What does it mean?
What do you mean by effective mass of earth and what is m ?
By complete I meant that$2\pi \sqrt {l/g}$ was a result of that.  Its derivation can be seen on :
http://www.quantumstudy.com/physics/simple-harmonic-motion-12/
@koolman  the concept I used is same as when we calculate acceleration due to gravity within the earth.
A better explanation is here http://physics.stackexchange.com/questions/99117/why-gravity-decreases-as-we-go-down-into-the-earth