Effect of Coriolis Force on a Particles Moving Horizontally on Earth

Let us assume that a particle of mass \(m\) is moving horizontally on earth with a velocity \(\vec{v}'\) at the place P whose angle of latitude is \(\lambda\). Draw a reference frame S' at point P whose X'-axis is towards east, Y'-axis is towards north and Z'-axis is vertically upwards. This reference frame is rotating with an angular velocity \(\vec{\omega}\) as earth is rotating about its north-south axis. Therefore,

\[ \boxed{ \vec{\omega} = \hat{j}'\omega\cos\lambda + \hat{k}'\omega\sin\lambda } \qquad ...(1) \]

and the velocity of particle is

\[ \boxed{ \vec{v}' = \hat{i}'v_x' + \hat{j}'v_y' } \qquad ...(2) \]

Thus the Coriolis force

\[ \vec{F}_c = -2m \left( \vec{\omega}\times\vec{v}' \right) \]
\[ = -2m \begin{vmatrix} \hat{i}' & \hat{j}' & \hat{k}'\\ 0 & \omega\cos\lambda & \omega\sin\lambda\\ v_x' & v_y' & 0 \end{vmatrix} \]
\[ \boxed{ \vec{F}_c = \hat{i}'\,2m\omega v_y'\sin\lambda - \hat{j}'\,2m\omega v_x'\sin\lambda + \hat{k}'\,2m\omega v_x'\cos\lambda } \qquad ...(3) \]

Thus horizontal component of Coriolis force is

\[ \boxed{ \vec{F}_{ch} = 2m\omega \left( \hat{i}'v_y' - \hat{j}'v_x' \right) \sin\lambda } \qquad ...(4) \]

and the magnitude of horizontal component of Coriolis force is

\[ \left|\vec{F}_{ch}\right| = 2m\omega\sin\lambda \sqrt{ v_x'^2+v_y'^2 } \]
\[ \boxed{ \left|\vec{F}_{ch}\right| = 2m\omega v'\sin\lambda } \qquad ...(5) \]

Thus the horizontal component of Coriolis force acts towards right in northern hemisphere and towards left in southern hemisphere.

The vertical component of Coriolis force is

\[ \boxed{ \vec{F}_c = \hat{k}'\,2m\omega v_x'\cos\lambda } \]

The vertical component of Coriolis force always acts upwards in both hemispheres.

Natural erosion of river banks, cyclones, etc. can be explained by the effect of Coriolis force on particles moving on the earth.

(i) Erosion of river banks

Suppose that water is flowing with a velocity \(\vec{v}=\hat{i}'v_x'\) from west to east in the river in northern hemisphere.

The horizontal component of Coriolis force acting on the water particles—

\[ \boxed{ \vec{F}_{ch} = -\hat{j}'\,2m\omega v_x'\sin\lambda } \]

It is evident from this equation that erosion takes place on the right bank of the river flowing from west to east in northern hemisphere due to Coriolis force and left bank of the river in the southern hemisphere.

(ii) Cyclones

Some times cyclone is generated in the atmosphere due to the inward spiraling motion of the air. The Cyclones can be explained on the basis of Coriolis force. When air of the atmosphere becomes hot at certain place, it goes up and creates low pressure at that place. As a result air rushes with a very high speed from high pressure area toward low pressure area. Hence Coriolis force starts acting on the air particles which rotate the air anti-clockwise spiraling upward in the northern hemisphere and clockwise in the southern hemisphere of the earth. This generates the cyclone.