What is the percentage change in the value of `g` as we shift from equator to pole on the surface of earth ? (Given equatorial radius of earth is greater than polar radius by `21 km` and mean radius of earth is `6300 km`).

Options

A. `0.52%`
B. `0.67%`
C. `1.67%`
D. `6.7%`

Explanation

To find the percentage change in the value of `g` as we move from the equator to the pole on the surface of the Earth, we can follow these steps:

### Step 1: Understand the formula for gravitational acceleration
The gravitational acceleration `g` at a distance `r` from the center of the Earth is given by the formula:
\[ g = \frac{GM}{r^2} \]
where `G` is the universal gravitational constant and `M` is the mass of the Earth.

### Step 2: Differentiate `g` with respect to `r`
To find how `g` changes with a change in `r`, we differentiate `g`:
\[ \frac{dg}{dr} = -\frac{2GM}{r^3} \]

### Step 3: Express the change in `g` as a percentage
The relative change in `g` can be expressed as:
\[ \frac{\Delta g}{g} \times 100 = -\frac{\Delta r}{r} \times 100 \]
where `\Delta r` is the change in radius.

### Step 4: Identify the values
Given:
- The difference in radius between the equator and the pole, `Δr = 21 km = 21 \times 10^3 m`
- The mean radius of the Earth, `r = 6300 km = 6300 \times 10^3 m`

### Step 5: Substitute the values into the formula
Now, substituting the values into the percentage change formula:
\[ \frac{\Delta g}{g} \times 100 = -\frac{21 \times 10^3}{6300 \times 10^3} \times 100 \]

### Step 6: Simplify the expression
Calculating the fraction:
\[ \frac{21}{6300} = \frac{1}{300} \]
Thus:
\[ \frac{\Delta g}{g} \times 100 = -\frac{2}{3} \times 100 \]

### Step 7: Calculate the percentage change
Calculating the percentage:
\[ -\frac{2}{3} \approx -0.67\% \]

### Conclusion
The percentage change in the value of `g` as we shift from the equator to the pole is approximately **-0.67%**, indicating a decrease in gravitational acceleration.

---