Which of the following are correct?

Options

A. If `R` is the radius of a planet, `g` is the acceleration due to gravity, the mean density of the planet is `3g//4piGR`
B. Acceleration due to gravity is a universal constant.
C. The escape velocity of a body from earth is `11.2 km s^(-1)`. The escape velocity from a planet which has double the mass of earth and half its radius is `22.4 km s^(-1)`
D. The ratio of gravitational mass and inertial mass of a body at the surface of earth is `1`.

Explanation

To determine which of the given statements are correct, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Statement**: "r is the radius of the planet, g is the acceleration due to gravity, mean density of the planet is \( \frac{3g}{4\pi g r} \), acceleration due to gravity is a universal constant."

1. We know that the acceleration due to gravity \( g \) at the surface of a planet is given by the formula:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( r \) is its radius.

2. The mean density \( \rho \) of the planet can be expressed in terms of mass and volume:
\[ \rho = \frac{M}{V} = \frac{M}{\frac{4}{3}\pi r^3} \]

3. Substituting \( M \) from the first equation into the density equation, we have:
\[ \rho = \frac{g r^2}{G \cdot \frac{4}{3}\pi r^3} = \frac{3g}{4\pi G r} \]
This means the expression for mean density given in the option is incorrect because it has \( g \) in the numerator instead of \( G \).

4. The statement that "acceleration due to gravity is a universal constant" is also false. It varies depending on the mass and radius of the planet.

**Conclusion for Option 1**: Incorrect.

### Step 2: Analyze Option 2
**Statement**: "Acceleration due to gravity is a universal constant."

1. As derived in the previous step, acceleration due to gravity \( g \) depends on the mass and radius of the planet:
\[ g = \frac{GM}{r^2} \]
Since \( M \) and \( r \) vary from one planet to another, \( g \) is not a universal constant.

**Conclusion for Option 2**: Incorrect.

### Step 3: Analyze Option 3
**Statement**: "Escape velocity of a body from Earth is 11.2 km/s, escape velocity from a planet who has double the mass of Earth and half its radius is 22.4 km/s."

1. The escape velocity \( v_e \) is given by:
\[ v_e = \sqrt{\frac{2GM}{r}} \]

2. For Earth, we have:
\[ v_{e,\text{Earth}} = 11.2 \text{ km/s} \]

3. For a planet with double the mass \( (2M) \) and half the radius \( \left(\frac{r}{2}\right) \):
\[ v_{e,\text{planet}} = \sqrt{\frac{2G(2M)}{\frac{r}{2}}} = \sqrt{\frac{4GM}{r}} = 2\sqrt{\frac{2GM}{r}} = 2v_{e,\text{Earth}} = 2 \times 11.2 \text{ km/s} = 22.4 \text{ km/s} \]

**Conclusion for Option 3**: Correct.

### Step 4: Analyze Option 4
**Statement**: "The ratio of gravitational mass and inertial mass of a body at the surface is 1."

1. This statement is based on the equivalence principle, which states that gravitational mass (the mass that determines the strength of the gravitational force) and inertial mass (the mass that determines how much an object resists acceleration) are equivalent.

**Conclusion for Option 4**: Correct.

### Final Conclusion
- **Correct Options**: Option 3 and Option 4.
- **Incorrect Options**: Option 1 and Option 2.