Which of the following statements are true about acceleration due to gravity?

Options

A. `g` decreases in moving away from the centre if `r gt R`.
B. `g` decreases in moving away from the centre if `r lt R`
C. `g` is zero at the centre of earth
D. `g` decreases if earth stops rotating on its axis.

Explanation

To determine which statements about acceleration due to gravity (g) are true, let's analyze each option step by step.

### Step 1: Understanding the Concept of Acceleration Due to Gravity
The acceleration due to gravity (g) is the force of gravity acting on an object per unit mass. It varies with distance from the center of the Earth.

### Step 2: Analyzing Option 1
**Statement 1:** "G decreases in moving away from center if R is greater than Capital R."

- When we move away from the Earth (beyond its surface), the formula for gravitational acceleration is given by:
\[ g' = g \frac{R^2}{(R + h)^2} \]
where \( R \) is the radius of the Earth, \( h \) is the height above the Earth's surface, and \( g \) is the acceleration due to gravity at the surface.
- As \( h \) increases (moving away from the Earth), \( (R + h)^2 \) increases, causing \( g' \) to decrease.
- **Conclusion:** This statement is true.

### Step 3: Analyzing Option 2
**Statement 2:** "G decreases in moving away from the center if Capital R is greater than small r."

- When moving inside the Earth (at a depth), the formula for gravitational acceleration is:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( d \) is the depth below the surface.
- As we move towards the center of the Earth, \( d \) increases, which means \( g' \) decreases.
- However, if we are moving away from the center (towards the surface), \( d \) decreases, leading to an increase in \( g' \).
- **Conclusion:** This statement is false.

### Step 4: Analyzing Option 3
**Statement 3:** "G is zero at the center of Earth."

- At the center of the Earth, the gravitational forces from all directions cancel out, resulting in no net gravitational force.
- Therefore, \( g \) is indeed zero at the center.
- **Conclusion:** This statement is true.

### Step 5: Analyzing Option 4
**Statement 4:** "G decreases if Earth stops rotating about its axis."

- The effective gravitational acceleration is given by:
\[ g' = g - r \omega^2 \cos^2 \theta \]
where \( \omega \) is the angular velocity of the Earth's rotation.
- If the Earth stops rotating, the term \( r \omega^2 \cos^2 \theta \) becomes zero, meaning \( g' \) would increase.
- **Conclusion:** This statement is false.

### Final Summary of True Statements
- **True Statements:** Option 1 and Option 3.
- **False Statements:** Option 2 and Option 4.