GATE-2018 (24) :Wyllie – Time average equation

1. The difference in the Earth’s gravity fields at the poles and the equator is at present 5.186 gals. If the velocity of the Earth’s rotation is reduced by 25% of its present value, then the difference in the gravity fields would be around (in gals)

1. 2.54

2. 3.71

3. 3.81

4. 4.30

Solution:

 

The relation between Earth’s rotation and gravity field is:

Rotation of the earth is 463 m/sec

If earth rotation is reduced to 25% then its rotation is 347.23m/sec

Then 463m/sec ----5.186gals

 

       347 m/sec ----X

 

X= (5.186*347) / 463

 Gravity (X) =  3.88 gals

-----------------------------------Another solution---------------------------------------------------

(Special Thanks to Pawan Singh)

The reduction of the centrifugal acceleration is largest at the equator and zero at poles. Due to this poleward increase of gravity, amounting to about 3375 mgal.

From the above information,

At present Centrifugal acceleration is equal to 3375 mgal

$a_R=R \omega^2=3375 mgal$

when the Earth rotation is slowed by 25%

$a^1_R=R \left(\frac{3}{4} \omega\right)^2$

$a^1_R=\frac{9}{16} R \omega^2$

$a^1_R=\frac{9}{16}a_R$

acceleration is $\frac{9}{16} a_R$

so, Change is acceleration$ =1a_R-\frac{9}{16}a_R= \frac{7}{16}a_R$

Decrease in acceleration =$ \frac{7}{16} \times 3375$

$=1476.5 mgal$

$=1.476 gal$

So, Net gravity difference= $5.186-1.476 gal$

$=3.709 gal$

$=3.71 gal$

Reference: Fundamentals of Geophysics, William Lowrie.

Extra information:

Calculating velocity of Earth’s rotation

V =D/T 

 

D = Circumference Of Earth 

 

Formula for circumference of earth = 2πr

                                                = 2*3.142*6371

                                                =40035.364 km

                                                          = 40035.364*1000

                                                = 40035364m 

 

Time taken for Earth to rotate around itself = 

 24 hours = 24*60*60

T = 86400 seconds

V= D/T 

 

= 40035364/86400

= 463.372

Velocity of earth rotation V = 463 m/sec