If the mass of a planet is `10%` less than that of the earth and the radius is `20%` greater than that of the earth, the acceleration due to gravity on the planet will be

Options

A. `5/8` times that on the surface of the earth
B. `3/4` times that on the surface of the earth
C. `1/2` times that on the surface of the earth
D. `9/10` times that on the surface of the earth

Explanation

To find the acceleration due to gravity on a planet with a mass that is 10% less than that of Earth and a radius that is 20% greater than that of Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for acceleration due to gravity (g)**:
The formula for acceleration due to gravity is given by:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Determine the mass of the planet**:
Given that the mass of the planet is 10% less than that of Earth, we can express the mass of the planet (\( M_p \)) as:
\[ M_p = M_e - 0.1 \cdot M_e = 0.9 \cdot M_e \]
where \( M_e \) is the mass of Earth.

3. **Determine the radius of the planet**:
The radius of the planet is 20% greater than that of Earth. Therefore, we can express the radius of the planet (\( R_p \)) as:
\[ R_p = R_e + 0.2 \cdot R_e = 1.2 \cdot R_e \]
where \( R_e \) is the radius of Earth.

4. **Substitute the values into the formula for g**:
Now we can substitute \( M_p \) and \( R_p \) into the formula for acceleration due to gravity:
\[ g' = \frac{G \cdot M_p}{R_p^2} = \frac{G \cdot (0.9 \cdot M_e)}{(1.2 \cdot R_e)^2} \]

5. **Simplify the equation**:
Simplifying the denominator:
\[ (1.2 \cdot R_e)^2 = 1.44 \cdot R_e^2 \]
Thus, we have:
\[ g' = \frac{G \cdot (0.9 \cdot M_e)}{1.44 \cdot R_e^2} \]

6. **Factor out the constants**:
We can express \( g' \) in terms of \( g \) (acceleration due to gravity on Earth):
\[ g' = \frac{0.9}{1.44} \cdot \frac{G \cdot M_e}{R_e^2} = \frac{0.9}{1.44} \cdot g \]

7. **Calculate the fraction**:
Now, we calculate \( \frac{0.9}{1.44} \):
\[ \frac{0.9}{1.44} = \frac{9}{14.4} = \frac{5}{8} \]
Therefore, we have:
\[ g' = \frac{5}{8} \cdot g \]

### Final Result:
The acceleration due to gravity on the planet is:
\[ g' = \frac{5}{8} \cdot g \]