Hint 1

(a) This is the square of the 4-momentum. It is equal to $-m^2c^2$.

(b) This is the centrifugal force.

(c)

$${\bf F}_{\rm Cor}=-2m\,\hbox{\boldmath $\omega$}\times{\bf v}_r$$

(d) At a distance $r$ from the centre of the Earth only the mass inside a sphere of radius $r$ contributes to the gravitational field. This mass is proportional to $r^3$. Combining this with the inverse-square law for the gravitational field gives force that is proportional to $r$.

(e) Your diagram should show the normal to the plate and two axes parallel to the sides of the rectangle (all passing through its centre).