Which of the following options are correct ?

Options

A. Acceleration due to gravity decrease with icreasing altitude
B. Acceleration due to gravity increase with increasing depth (assume the earth to be a sphere of uniform density)
C. Acceleration due to gravity increase with increasing latitude
D. Acceleration due to gravity is independent of the mass of the earth

Explanation

To solve the question regarding which of the options about acceleration due to gravity are correct, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Option 1: Acceleration due to gravity decreases with increasing altitude.**
- The formula for acceleration due to gravity at a height \( H \) above the Earth's surface is given by:
\[ g' = \frac{g R^2}{(R + H)^2} \]
- Here, \( g \) is the acceleration due to gravity at the surface, \( R \) is the radius of the Earth, and \( H \) is the height above the surface.
- As \( H \) increases, the denominator \( (R + H)^2 \) increases, which causes \( g' \) to decrease.
- **Conclusion:** This statement is **correct**.

2. **Option 2: Acceleration due to gravity increases with increasing depth.**
- The formula for acceleration due to gravity at a depth \( d \) inside the Earth (assuming uniform density) is:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
- As \( d \) increases, the term \( \frac{d}{R} \) increases, which means \( g' \) decreases.
- **Conclusion:** This statement is **incorrect**.

3. **Option 3: Acceleration due to gravity increases with increasing latitude.**
- The formula for acceleration due to gravity at latitude \( \lambda \) is:
\[ g' = g - \omega^2 R \cos^2(\lambda) \]
- Here, \( \omega \) is the angular velocity of the Earth and \( R \) is the radius of the Earth.
- As latitude \( \lambda \) increases, \( \cos^2(\lambda) \) decreases, which means the term \( \omega^2 R \cos^2(\lambda) \) decreases, leading to an increase in \( g' \).
- **Conclusion:** This statement is **correct**.

4. **Option 4: Acceleration due to gravity is independent of the mass of the Earth.**
- The formula for acceleration due to gravity is:
\[ g = \frac{GM}{R^2} \]
- Here, \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth. This clearly shows that \( g \) is directly proportional to the mass of the Earth.
- **Conclusion:** This statement is **incorrect**.

### Final Answer:
The correct options are **1 and 3**.