Weight of a body depends directly upon acceleration due to gravity `g`. Value of `g` depends upon many factors. It depends upon the shape of earth, rotation earth etc. Weight of a body at a pole is more then that at a place on equator because g is maximum at poles and minimum on equator. Acceleration due to gravity g varies with latitude `lambda` as per relation given below :
`g_(rot)=g-Romega^(2)cos^(2)lambda` where `R` is radius of earth and `omega ` is angular velocity of earth. A body of mass `m` weighs `W_(r )` in a train at rest. The train then begins to run with a velocity v around the equator from west to east. It observed that weight `W_(m)` of the same body in the moving train is different from `W_(r )`. Let `v_(e )`be the velocity of a point on equator with respect to axis of rotation of earth and R be the radius of the earth. Clearly the relative between earth and trainwill affect the weight of the body.
Weight `W_(m)` of the body can be given as

Options

A. `mg-m((v_(e)+v)^(2))/(R )`
B. `mg-m((v_(e)-v)^(2))/(R )`
C. `(m)/(R )[v_(e )^(2)-(v_(e )+v)^(2)]`
D. `mg+m((v_(e)+v)^(2))/(R )`

Explanation

To solve the problem, we need to analyze how the weight of a body changes when it is in a moving train. The weight of the body at rest is given as \( W_r \). When the train starts moving, the effective weight \( W_m \) of the body will change due to the additional centrifugal force acting on it because of the motion of the train.

### Step-by-Step Solution:

1. **Understanding Weight at Rest**:
The weight of the body at rest in the train is given by:
\[
W_r = m g_{rot}
...