Effects of Centrifugal Forces on Earth

Let a particle is situated at a point P on the earth whose latitude is \(\lambda\). Assuming point P as origin draw a reference frame whose X' axis is towards east, Y' axis is towards north and Z' axis is vertically upwards as shown in Fig.(1). This reference frame rotates with an angular velocity \(\vec{\omega}\) due to the rotation of the earth.

If unit vectors along the axes of reference frame are \(\hat{i}'\), \(\hat{j}'\), \(\hat{k}'\), then

\[ \vec{\omega} =\hat{i}'\omega\cos (90)+ \hat{j}'\omega\cos(\lambda) + \hat{k}'\omega\cos(90-\lambda) \qquad ...(1) \]

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

If the particle is at rest on the earth, then Coriolis force will be zero and only centrifugal force will appear to be acting on the particle.

Let actual acceleration due to gravity at point P is \(a=g\) and apparent acceleration due to gravity due to rotational motion of earth is \(a'=g_\lambda\).

Hence,

\[ \boxed{ \vec{g}_{\lambda} = \vec{g} - \vec{\omega}\times \left( \vec{\omega}\times\vec{r} \right) } \qquad ...(2) \]

where vector \(\vec{r}\) is a position vector of point P about centre of rotation O'.

\[ \vec{r} =\hat{i}'r\cos (90)+ \hat{j}'r\cos(90+\lambda) + \hat{k}'\omega\cos(\lambda) \]

\[ \boxed{ \vec{r} = -\hat{j}'r\sin\lambda + \hat{k}'r\cos\lambda } \]

If radius of earth is \(R\), then \(r=R\cos\lambda\). (in \(\Delta\) OO'P)

\[ \boxed{ \vec{r} = -\hat{j}'R\sin\lambda\cos\lambda + \hat{k}'R\cos^2\lambda } \qquad ...(3) \]

Substituting equations (1) and (3) in equation (2), we get

\[ \vec{g}_{\lambda} = -\hat{k}'g - \vec{\omega} \times \left[ \left( \hat{j}'\omega\cos\lambda + \hat{k}'\omega\sin\lambda \right) \times \left( -\hat{j}'R\sin\lambda\cos\lambda + \hat{k}'R\cos^2\lambda \right) \right] \]
\[ \vec{g}_{\lambda} = -\hat{k}'g - \vec{\omega} \times \left( \hat{i}'\omega R\cos\lambda \right) \]
\[ \vec{g}_{\lambda} = -\hat{k}'g - \left( \hat{j}'\omega\cos\lambda + \hat{k}'\omega\sin\lambda \right) \times \left( \hat{i}'\omega R\cos\lambda \right) \]
\[ \text{or}\qquad \vec{g}_{\lambda} = -\hat{k}'g + \hat{k}'\omega^2R\cos^2\lambda - \hat{j}'\omega^2R\cos\lambda\sin\lambda \]
\[ \boxed{ \vec{g}_{\lambda} = -\hat{k}' \left( g-\omega^2R\cos^2\lambda \right) - \hat{j}'\omega^2R\cos\lambda\sin\lambda } \qquad ...(4) \]

∴ Magnitude of effective \(g\)

\[ \left|\vec{g}_{\lambda}\right| = \sqrt{ \left( g-\omega^2R\cos^2\lambda \right)^2 + \left( \omega^2R\cos\lambda\sin\lambda \right)^2 } \]

Since \(\omega\) is a small quantity, so the term containing \(\omega^4\) can be neglected. Hence

\[ \boxed{ \left|\vec{g}_{\lambda}\right| = g-\omega^2R\cos^2\lambda } \qquad ...(5) \]

This equation (5) describes the effect of rotation of earth on acceleration due to gravity. If the direction of observed acceleration due to gravity makes an angle \(\theta\) with the direction of real acceleration due to gravity, then

\[ \tan\theta = \frac{ \omega^2R\cos\lambda\sin\lambda }{ g-\omega^2R\cos^2\lambda } \]

∵ \(\omega^2R\cos^2\lambda\) is negligible in comparison to \(g\), so it can be neglected. Hence

\[ \boxed{ \tan\theta = \frac{\omega^2R\cos\lambda\sin\lambda}{g} = \frac{\omega^2R}{2g}\sin2\lambda } \qquad ...(6) \]

Case (i) At pole of the earth :

\[ \lambda=90^\circ,\qquad\cos\lambda=0 \] \[ \therefore\quad g_p=g \]

Thus the acceleration due to gravity at poles is equal to the real acceleration due to gravity and its direction is towards the centre of the earth.

Case (ii) At equator :

\[ \lambda=0,\qquad\cos\lambda=1 \] \[ \therefore\quad g_e=g-\omega^2R \] \[ \text{and}\qquad \theta_0=0 \]

Thus at equator there is maximum effect of rotation of earth on acceleration due to gravity and the direction of acceleration due to gravity in this case also towards the centre of the earth.

(iii) At \(\lambda=45^\circ\)

\[ g_{\lambda} = g-\frac{\omega^2R}{2} \] \[ \text{and}\qquad \theta = \tan^{-1} \left( \frac{\omega^2R}{2g} \right) \]

Thus at \(\lambda=45^\circ\) the direction of observed acceleration due to gravity is inclined maximum towards the direction of the real acceleration due to gravity.