Variation of g — Revision Notes

⚡ 30-Second Revision

Surface 'g' — $g = G M / R^2$
Altitude 'h' — $g_h = g(1 + h/R)^{-2}$
Altitude (approx. for $h ll R$) — $g_h \approx g(1 - 2h/R)$
Depth 'd' — $g_d = g(1 - d/R)$
Center of Earth ($d=R$) — $g_d = 0$
Latitude $\lambda$ — $g' = g - R\omega^2 \cos^2\lambda$
Equator ($\lambda=0^circ$) — $g'_{equator} = g - R\omega^2$ (minimum 'g' due to rotation)
Poles ($\lambda=90^circ$) — $g'_{poles} = g$ (maximum 'g' due to rotation)

2-Minute Revision

The acceleration due to gravity, 'g', is not constant. It varies primarily with altitude, depth, and latitude. With increasing altitude ($h$), 'g' decreases because you are further from the Earth's center.

The exact formula is $g_h = g(1 + h/R)^{-2}$, which simplifies to $g_h approx g(1 - 2h/R)$ for small heights. With increasing depth ($d$), 'g' also decreases, becoming zero at the Earth's center. The formula is $g_d = g(1 - d/R)$.

This happens because only the mass of the Earth within the sphere of radius $(R-d)$ contributes to the gravitational pull. Finally, due to Earth's rotation, 'g' is minimum at the equator and maximum at the poles.

The effective 'g' at latitude $lambda$ is $g' = g - Romega^2 cos^2lambda$. Remember that 'g' decreases twice as fast with height as it does with depth for small changes. Mastering these formulas and their conceptual basis is key for NEET.

5-Minute Revision

A thorough revision of 'Variation of g' involves understanding its three main causes: altitude, depth, and Earth's rotation. At the Earth's surface, $g = GM/R^2$.

1
Variation with Altitude (Height $h$): — As an object moves to a height $h$ above the surface, its distance from the Earth's center becomes $R+h$. The exact formula is g_h = g left(\frac{R}{R+h}\right)^2 = g left(1 + \frac{h}{R}\right)^{-2}. For small heights ($h ll R$), we use the binomial approximation: g_h approx g left(1 - \frac{2h}{R}\right). This shows 'g' decreases with height. For example, if $h = R/100$, $g_h approx g(1 - 2/100) = 0.98g$, a 2% decrease.

1
Variation with Depth (Depth $d$): — When an object is at depth $d$ below the surface, only the mass of the Earth within a sphere of radius $(R-d)$ contributes to the gravitational force. Assuming uniform density, the formula is g_d = g left(1 - \frac{d}{R}\right). 'g' decreases linearly with depth and becomes zero at the Earth's center ($d=R$). For instance, at $d=R/2$, $g_d = g(1 - 1/2) = g/2$.

1
Variation with Latitude ($lambda$): — Earth's rotation causes an outward centrifugal effect, reducing the effective 'g'. This effect is maximum at the equator ($lambda=0^circ$) and zero at the poles ($lambda=90^circ$). The effective 'g' at latitude $lambda$ is $g' = g - Romega^2 cos^2lambda$. Thus, $g'_{equator} = g - Romega^2$ (minimum) and $g'_{poles} = g$ (maximum). The difference $Romega^2$ is approximately $0.034,\text{m/s}^2$.

Key takeaway: 'g' is maximum at the poles and on the surface, and zero at the Earth's center and infinitely far away. For small changes, 'g' decreases twice as fast with height as with depth. Practice applying these formulas to numerical problems and understanding the conceptual implications.

Prelims Revision Notes

Variation of 'g' for NEET UG Physics

1. Fundamental Definition:

Acceleration due to gravity ($g$) is the acceleration experienced by an object due to Earth's gravitational pull.
Standard value at Earth's surface: $9.8,\text{m/s}^2$.
Formula: $g = G M / R^2$, where $G$ is universal gravitational constant, $M$ is Earth's mass, $R$ is Earth's radius.

2. Variation with Altitude (Height $h$):

Reason: — Increased distance from Earth's center.
Exact Formula: — g_h = g left(\frac{R}{R+h}\right)^2 = g \left(1 + \frac{h}{R}\right)^{-2}
Approximate Formula (for $h \ll R$, typically $h < 0.05R$): — $g_h \approx g \left(1 - \frac{2h}{R}\right)$
Observation: — 'g' decreases as height increases. At $h \to \infty$, $g_h \to 0$.
Rate of decrease: — For small $h$, the decrease is approximately $2gh/R$.

3. Variation with Depth (Depth $d$):

Reason: — Only the mass of Earth within the sphere of radius $(R-d)$ contributes to gravity.
Formula (assuming uniform density): — $g_d = g \left(1 - \frac{d}{R}\right)$
Observation: — 'g' decreases linearly as depth increases.
At Earth's center ($d=R$): — $g_d = 0$.
Rate of decrease: — For small $d$, the decrease is approximately $gd/R$.

4. Variation with Latitude (Due to Earth's Rotation):

Reason: — Centrifugal effect due to Earth's rotation opposes gravity.
Formula: — $g' = g - R\omega^2 \cos^2\lambda$

* $g'$: Effective 'g' at latitude $\lambda$. * $g$: 'g' if Earth were non-rotating. * $R$: Earth's radius. * $\omega$: Earth's angular velocity. * $\lambda$: Latitude.

At Equator ($\lambda=0^circ$, $\cos\lambda=1$): — $g'_{equator} = g - R\omega^2$ (Minimum value of 'g' due to rotation).
At Poles ($\lambda=90^circ$, $\cos\lambda=0$): — $g'_{poles} = g$ (Maximum value of 'g' due to rotation).
Observation: — 'g' is maximum at poles and minimum at equator due to rotation.

5. Key Comparisons & Points:

For small changes, 'g' decreases twice as fast with height as with depth ($2h/R$ vs $d/R$).
'g' is maximum at the surface (specifically, at the poles on the surface).
'g' is zero at the Earth's center and at infinite distance.
Earth's non-spherical shape (oblate spheroid) also contributes to 'g' being slightly higher at poles than equator, even without considering rotation, due to poles being closer to the center.

Vyyuha Quick Recall

All Deep Layers Rotate: Altitude, Depth, Latitude, Rotation.

Altitude: $g$ goes Away (decreases).
Depth: $g$ goes Down (decreases to zero).
Latitude: $g$ is Less at the equator, Larger at poles.
Rotation: Reduces $g$ (except at poles).