I need help with these problems they must be done with dimen

I need help with these problems they must be done with dimensional analysis.. please help

A hole is drilled through the center of the Earth along a diameter. If a rock is then dropped into this hole at one end, it will oscillate back and forth between the two ends of the hole. Use dimensional analysis to find an expression for the period of these oscillations. A rod is hinged to the follr at one end. It is held straight up and then released at rest. Due to a small kick an air molecule, the rod begins fallig over. Use dimensional analysis to find an expression for angular velocity Ws with which it hits the floor. A small particle is released at rest and begins falling through a fluid. Use dimensional analysis to find an expression for its terminal velocity in each of the following owing two cases The fluid is a liquid, so the resistive force is given by F = a upsilon, where a is constant. The fluid is a gas, so the resistive force is given by F = a upsilon^2, where a is constant. How does the height that an animal can jump scale with its mass? By considering the sizes of bones and muscles, determine how the strength of animal scale with its mass. Which can lift a larger fraction of its own weight, a large animal or a small animal?

Solution

1) suppose particle is at distance r from centre

mass of earth below particle.

M = Me ( 4 pi r^3 / 3 ) / (4 pi R^3 / 3) = Me r^3 / R^3

Force, Fg = G M m / r^2

Fg = ( G Me r^3 m ) / (r^2 R^3 )

Fg = (G Me m / R^3) r

F is directly propotional to r so particle will perform SHM.

k = G Me m / R^3

Time period , T = 2 pi sqrt(m / k )

= 2pi sqrt[ m / (G Me m / R^3 ) ]

= 2pi sqrt[ R^3 / G Me ] ...............Ans

2. As rod will come to ground, its centre of mass will come down distance L/2 .

Applying energy conservation,

PEi + KEi   = PEf + KEf

(m g L /2) + 0 = 0 + ( I w^2 ) /2

m g L   = ( m L^2 / 3 ) w^2

w = sqrt[ 3 g / L ]