Variation of Acceleration Due to Gravity (g) due to Shape of Earth | Notes, Important Questions & Answers, MCQS, Quiz

Introduction

The acceleration due to gravity (g) varies due to multiple factors, one of which is the shape of the Earth. The Earth is not a perfect sphere; rather, it is an oblate spheroid, meaning it is slightly flattened at the poles and bulging at the equator. This shape influences the value of g at different locations.

The value of g changes even on the surface of the earth also. For example, g is the highest on the poles and lowest on the equator.

The earth is not perfectly spherical but is an oblate spheroid. The polar radius (radius near poles) of earth is 21 km smaller than its equatorial radius (near the equator). Earth is not spherical, but actually, it is bulged out, as shown below.

As per the formula derived, the acceleration due to gravity is inversely proportional to the square of the radius of the earth. The radius of the earth at the equator is more; at the equator, g is less. This is the reverse in the case of poles.

Suppose we consider the shape of the earth as slightly elliptical. So, we will have different distances from the pole and the equator from the center.

So, the distance between the pole (R_P) and the equator (R_E) from the center is as,

R_E > R_P

And, from observation, we have the relation between R_E and R_P as,

g ∝ 1/R²

Then, if we consider G and M as constants in the acceleration formula, then,

g_P ∝ 1/R_P² and 

g_E ∝ 1/R_E²

So, the gravitational accelerations on the equator and pole are given by,

g_P > g_E

Here, from the distances between the poles and the equator, we have the relationship of gravitational acceleration as, 

g_P / g_E = R_E² / R_P²

Hence, the gravitational acceleration at the equator is less than the gravitational acceleration at the pole. So if a person moves from the equator to the poles, their weight decreases as the value of g decreases.

Why Does g Vary Due to the Earth’s Shape?

1. Oblate Shape of the Earth
   • The Earth’s equatorial radius is greater than its polar radius.
   • Since g is inversely proportional to the square of the radius, it is lower at the equator and higher at the poles.
2. Centrifugal Force
   • The Earth’s rotation generates centrifugal force, which is maximum at the equator and zero at the poles.
   • This force reduces the effective value of g at the equator, further lowering its magnitude compared to the poles.

Mathematical Representation of Variation of g

• The standard value of g at sea level is 9.8 m/s².
• At the poles: g = 9.832 m/s² (maximum)
• At the equator: g = 9.780 m/s² (minimum)
• The approximate difference in g between the poles and the equator is 0.052 m/s².
• The formula incorporating the shape of the Earth: g′=g(1−(R_e²−R_p²)/R²)
  • where: g = effective acceleration due to gravity
  • R_e = equatorial radius
  • R_P = polar radius
  • R = Earth’s mean radius

FAQs on Variation of g Due to the Shape of the Earth

Q1. Why is g higher at the poles than at the equator?

A1. g is higher at the poles because the polar radius is smaller than the equatorial radius, leading to a stronger gravitational pull. Additionally, there is no centrifugal force at the poles.

Q2. How does Earth’s rotation affect g?

A2. The Earth’s rotation generates centrifugal force, which counteracts gravitational force. This effect is strongest at the equator, where it reduces g, and absent at the poles.

Q3. Can the variation in g due to shape affect objects on Earth?

A3. Yes, but the effect is very small for everyday activities. However, precise scientific measurements and satellite calibrations must consider this variation.

Q4. How much does g differ from the equator to the poles?

A4. The difference is approximately 0.052 m/s², with g being about 9.832 m/s² at the poles and 9.780 m/s² at the equator.

MCQs on Variation of g Due to Earth’s Shape

Q1. What is the primary reason for the variation of g due to the shape of the Earth?
A) Change in mass distribution
B) Earth’s magnetic field
C) Earth’s oblate shape
D) Presence of the atmosphere

Answer: C) Earth’s oblate shape
Explanation: The Earth is an oblate spheroid, meaning its equatorial radius is larger than its polar radius. Since g is inversely proportional to the square of the radius, its value decreases at the equator and increases at the poles.

Q2. Where is the acceleration due to gravity maximum on Earth?
A) At the equator
B) At the poles
C) At the center of the Earth
D) At sea level

Answer: B) At the poles
Explanation: Due to the smaller radius and absence of centrifugal force, g is maximum at the poles.

Q3. How does the centrifugal force affect the value of g at the equator?
A) Increases g
B) Decreases g
C) Has no effect
D) Depends on the latitude

Answer: B) Decreases g
Explanation: The centrifugal force acts outward and counteracts gravity, reducing the effective value of g at the equator.

Test Your Knowledge

Conclusion

The variation of g due to the Earth’s shape plays an essential role in geophysics, engineering, and astronomy. Although the change is small, it has significant implications in satellite mechanics, GPS accuracy, and scientific experiments.

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