Suppose a planet exists whose mass and radius both, are half those of earth. Calculate the acceleration due to gravity on the surface of this planet.

Explanation

To calculate the acceleration due to gravity on the surface of a planet whose mass and radius are both half that of Earth, we can follow these steps:

### Step 1: Understand the Formula for Acceleration Due to Gravity
The formula for acceleration due to gravity (g) at the surface of a planet is given by:
\[ g = \frac{GM}{R^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 2: Define the Mass and Radius of the New Planet
Let:
- The mass of Earth be \( M \),
- The radius of Earth be \( R \).

For the new planet:
- Mass \( M_p = \frac{M}{2} \) (half of Earth's mass),
- Radius \( R_p = \frac{R}{2} \) (half of Earth's radius).

### Step 3: Substitute the Values into the Formula
Now, we can substitute the values of mass and radius of the new planet into the formula for acceleration due to gravity:
\[ g_p = \frac{G \left(\frac{M}{2}\right)}{\left(\frac{R}{2}\right)^2} \]

### Step 4: Simplify the Expression
Now, simplify the expression:
\[ g_p = \frac{G \left(\frac{M}{2}\right)}{\frac{R^2}{4}} \]
\[ g_p = \frac{G M}{2} \times \frac{4}{R^2} \]
\[ g_p = \frac{4GM}{2R^2} \]
\[ g_p = \frac{2GM}{R^2} \]

### Step 5: Relate it to Earth's Gravity
We know that the acceleration due to gravity on Earth is:
\[ g_e = \frac{GM}{R^2} \]
Thus, we can express \( g_p \) in terms of \( g_e \):
\[ g_p = 2g_e \]

### Step 6: Substitute the Value of Earth's Gravity
The average acceleration due to gravity on Earth is approximately \( 9.8 \, \text{m/s}^2 \):
\[ g_p = 2 \times 9.8 \, \text{m/s}^2 \]
\[ g_p = 19.6 \, \text{m/s}^2 \]

### Conclusion
The acceleration due to gravity on the surface of the planet is:
\[ g_p = 19.6 \, \text{m/s}^2 \]

---