Question on gravitation and rotation of the Earth

[AakashPandita]

Homework Statement

A body is suspended on a spring balance in a ship sailing along the equator with a speed $v'$ . If $\omega$ is the angular speed of the Earth and $\omega_0$ is the scale reading when the ship is at rest , the scale reading when the ship is sailing, will be very close to
(a) $\omega_0$
(b) $(\omega_0)(1 + \frac { 2 \omega v'}{g})$
(c) $(\omega_0)(1 \mp \frac { 2 \omega v'}{g})$
(d)none of these

Homework Equations

Let mass of object be m
Radius of Earth be R.
Tension in spring balance=Reading in spring balance = $\omega_0$
Tension be T when ship not sailing and Tension be T' when ship is sailing

When ship is not sailing
$mg - T = m (\omega)^2 R$
When ship is sailing
$mg - T' = m (\omega \mp v'/R)^2 R$
I solved for T' but the answer is not coming.
I could show the working but latex is hard to write.

 

[haruspex]

I could show the working but latex is hard to write.

But if you don't post your working nobody can tell where you are going wrong.

 

[AakashPandita]

 

[haruspex]

That's fine so far, but notice that in the answers everything is expressed as a multiple of $\omega_0$. Get your answer into that form. (You will probably also need to discard some third order term.)

 

[AakashPandita]

Wait a minute. Doesn't the value of g that we use account for the centripetal force due to rotation of earth? Is $g- (\omega)^2 R =g$ using approximation? If yes then how?

 

[AakashPandita]

I got the answer. Thanks a lot.